in our research , we kept running across claims that the number one `` used to be prime . '' for example , at the time we wrote this article , the sequence http://oeis.org/a008578 [ ] in oeis was named `` prime numbers at the beginning of the 20th century ( today 1 is no longer regarded as a prime ) . '' and before we changed it in late 2011 , the english wikipedia article on _ prime number _ said `` until the 19th century , most mathematicians considered the number 1 a prime . '' there were ( and are ) individuals who defined the number one to be prime , but as we will see in this brief note , the history is much more complicated than `` one used to be prime . '' at first , most started the sequence of primes with 2 , but some began with 3 and a rare few began with 1. there does not appear to be any period of time during which most mathematicians defined one to be a prime .before we start , let s make two points clear .first , whether or not a number ( especially unity ) is a prime is a matter of definition , so a matter of choice , context and tradition , not a matter of proof .yet definitions are not made at random ; these choices are bound by our usage of mathematics and , especially in this case , by our notation .we will see , for example , that as the use of the integers changed , one ( unity , the monad ) , which at first was not generally viewed as a number by the ancient greeks , ` becomes ' a number some 2,000 years later . for much of historyit did not even make sense to ask if the number one was a prime .second , when we stick to the ( modern ) positive integers , it is not difficult to adjust our theorems to allow the number one to be called a prime .for example , some say unity is omitted from the primes to preserve the uniqueness of factorization. however , in the past , mathematicians who defined one to be prime just added a couple words to the fundamental theorem of arithmetic ( fta ) . if we start with the version of the fta stated by crandall and pomerance ) , we need only add the two characters `` '' :  for each natural number there is a unique factorization where exponents are positive integers and are primes .the real problem with one being a prime only became apparent in the 18th century when mathematicians expanded their purview to wider notions of integers , such as the gaussian integers and general number fields .in that context , they were forced to address the notion of units , to separate the concepts of irreducibility and primality , etc .the generalization of prime to unique factorization domains clarified the role of unity and now informs the way we define primality in the ordinary positive integers .to get started , we will show that in the early days of defining primes , which we will illustrate with euclid , the unit ( one ) was not even considered a number , so was _ ipso facto _ excluded from the primes .this begins to change when stevin showed how to calculate with decimal real numbers in 1585 , bringing on a confused period in which some held unity to be a prime and others did not . when gauss first states and proves the fundamental theorem of arithmetic ( fta ) in 1801 , writers begin to coalesce slowly around the modern definition of prime which again excludes unity .euclid , for example , defines unity ( ) , numbers ( ) , and the primes ( ) in book 7 of his _ elements _ as follows :  * a unit is that by virtue of which each of the things that exist is called one . *a number is a multitude composed of units . *a prime number is that which is measured by a unit alone .euclid ( and his contemporaries ) did not need to say explicitly that one was not a prime because primes were a subcategory of the numbers , so one was not a number .smith states it this way :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ aristotle , euclid , and theon of smyrna defined a prime number as a number `` measured by no number but an unit alone , '' with slight variations of wording . since unity was not considered a number , it was frequently not mentioned ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  writers at times broke this silence ; for example martianus capella ( c.400 , ) , an early and influential developer of the system of seven liberal arts , summarizes the properties of the first integers as follows :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we have briefly discussed the numbers comprising the first series , the deities assigned to them , and the virtue of each number .i shall now briefly indicate the nature of number itself , what relations numbers bear to each other , and what forms they represent . [ ] let us consider all numbers of the first series according to the above classifications : the monad is not a number ; the dyad is an even number ; the triad is a prime number , both in order and in properties ; the tetrad belongs in the even times even class ; the pentad is prime ; the hexad belongs to the odd times even or even times odd ( so it is called perfect ) ; the heptad is prime ; the octad belongs to [ ] _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  martinus and others such as nicomachus ( c.100 ) and iamblichus ( c.300 ) , boethius ( c.500 ) , and cassiodorus ( c.550 ) make the primes a subset of the odds , excluding both one and two from the primes so for them , the smallest prime was three .( it is easy to extend this list of those for whom the first prime was three well into the 16th century . )most of the ancient greeks however , like euclid , began the sequence of primes with two .for about 2000 years most writers varied little from the position of the ancient greeks .for example , figure [ fig_isadorus ] shows a copy of isidore of seville s ( c.636 ) work made in 1493 .    hereisidore ( ) writes `` number is a multitude made up of units . for oneis the seed of number but not number . ''nine centuries after isidore , j. kbel ( 1537 ) repeats this argument `` darauss verstehstu das 1 .kein zal ist / sonder es ist ein gebererin / anfang / und fundament aller anderer zalen '' ( see menninger ). simply put , for them numbers are multiples and therefore divisible , but unity is neither a multiple or divisible .( if you would like a longer argument , al - kind ( c.850 ) argues at great length that one is not a number , so that it is neither even nor odd . )omitting one from the numbers , hence the primes , was the general rule , but there were a few isolated exceptions .speusippus ( c.350bc ) , tarn writes ,  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is exceptional among pre - hellenistic thinkers in that he considers one to be the first prime number . and heath , _ hist .gr . math .. - , followed by ross , _ aristotle s physics _ , p. , and others , is mistaken when he contends that chrysippus , who is said to have defined one as > en ( cf .iamblichus , _ in nicom .p. ii , - [ pistelli ] ) , was the first to treat one as a number ( cf .further p. f. with note _ supra _ ) ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  a later exception may be rabbi ben ezra ( c.1140 ) in his _ sefer ha - echad _ .in 1585 simon stevin published his tract _ de thiende _ which laid the basis for modern decimal expansions . herestevin showed how to represent and operate on both the positive integers ( euclid s ) and the magnitudes ( lengths of geometrical objects , euclid s ) with the same notation and algorithms . in his thesis `` the concept of as a number , '' c. jones it this way :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in general , mathematics before stevin is of one character and , after him , it is of another reflecting his contributions . in this regard, he is like euclid : he stood at a watershed in the history of mathematics . andas with euclid , he was so successful that , from our present day vantage point , it is hard to see the other side of that watershed .over there , was not a number ; here and now , it is ; even is a number , and , and aleph null ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  no change happens instantly , and nearly a century later when moxon wrote the first english mathematical dictionary ( ) , moxon defined number ( with respect to unity ) with trepidation ( see figure [ fig_moxon ] ) .the argument moxon gives here , that 1 is a number otherwise was one of stevin s arguments , but it is best viewed as a metaphor for stevin s most convincing message : notationally and algorithmically , there is no difference between one and the other numbers .the redefinition of unity as a number now made it reasonable to ask if one was a prime ; however , the way primes were used in this period did not force any particular definition . in their history of the fundamental theorem of arithmetic , aargn and zkanwrite that at this time `` prime factorization was not looked upon as something of interest in its own right , but as a means of finding divisors '' .similarly , aargn and fletcher , writing about the existence and uniqueness components of the fta , note the following :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it is significant that propositions vii.31 and vii.30 of the elements lead immediately to their proofs although euclid forbears to take these steps .the first known proof of existence is due to al - farisi ( died c.1320 ) , but he did not go on to prove uniqueness , mainly because his interest was in the divisors of a number rather than the factorisation itself and if a mathematician s interest is in greatest common divisors , or perfect numbers , or amicable numbers then the divisors are the crucial objects , whereas the prime factorisation is just a means to an end ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  without clear direction on the primality of one , a period of confusion begins .many stuck with the tradition of excluding one from the primes , for example : p. a. cataldi ( 1603 ) , m. mersenne ( 1625 ) , lon de saint - jean ( 1657 ) , f. v. schooten ( 1657 ) , the shuli jingyun ( c.1720 ) and l. euler ( 1770 ) .however , it was also reasonable to define the primes as the complement of the set of composites , often called the incomposites , such as in the influential tables of brancker & pell ( 1688 ) .others who listed one as prime in this period are f. wallis ( 1685 ) , j. prestet ( 1689 ) , c. goldbach ( 1742 ) ( see figure [ fig_goldbach ] ) , j. h. lambert ( 1770 ) , a. felkel ( 1776 ) and e. waring ( 1782 ) .we could make these lists much longer ( see and ) , but these are sufficient to illustrate the point .keep in mind that these authors may , or may not , have stuck to the indicated view of the primality of one ; this list is just what they wrote at the indicated time .definitions depend on context ( as well as on tradition and usage ) , so for example , the later number theorist v. a. lebesgue ( 17911875 ) omitted unity from the primes in an 1856 article , included it in an 1859 article , and then excluded it in 1864 article . for many authors we must infer their position by their usage .for example , the snippet of goldbach s famous letter to euler in figure [ fig_goldbach ] about expressing integers as sums of prime makes his stand obvious , as does gauss counting 168 primes below 1,000 .when we look back on this period , it seems odd that factorizations were used without asking if those factorizations were unique .our modern notions of integers , primes and unique factorization appear so obvious to us that we are blinded by hindsight .gauss _ disquisitiones arithmeticae _ provides another watershed moment in the history of the integers .aargn and fletcher write   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it is not too much of an exaggeration to say that the result [ fta ] passed from being unknown to being obvious without a proof passing through the head of any mathematician .[ ] the first clear statement and proof of the fta seems to have been given by gauss ( 17751855 ) in his _ disquisitiones arithmeticae _ of 1801 ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  gauss not only stated fta ( and gave it a partial proof ) , but while doing so laid the ground work for making unique factorization central to our understanding of integers . equally important , he discussed generalizations of the integers , such as the gaussian integers , which made mathematicians look more carefully at the role of one ( and all the units ) in factorizations .gauss text never defines primes explicitly , but still provides an eloquent argument for the modern definition of prime .  even after gauss pivotal text , many continued to write that unity was prime . among theseare : a. m. legendre ( 1830 ) , e. hinkley ( 1853 ) , m. glaisher ( 1876 ) , k. weierstrass ( 1876 ) , r. frick & f. klein ( 1897 ) , a. cayley ( 1890 ) , l. kronecker ( 1901 ) , g. chrystal ( 1904 ) and d. n. lehmer ( 1914 ) .but on the other hand , we also have quite a number who omitted unity from the primes : g. s. klgel ( 1808 ) , p. barlow ( 1811 ) , m. ohm ( 1834 ) , a. reynaud ( 1835 ) , l. dirichlet ( 1863 ) , e. meissel ( 1870 ) , p. chebyshev ( 1889 ) , e. lucas ( 1891 ) , j. p. gram ( 1893 ) , e. landau ( 1909 ) and h. v. mangoldt ( 1912 ) .together these lists provide a strong argument against the contemporary legend that `` one used to be a prime number''the truth is far messier .another way we might assess the view of the primality of the number one near 1900 is by using the encyclopdia britannica .the encyclopdia britannica s 9th edition , published one volume at a time from 1875 to 1889 , is often called the scholar s edition because of its authenticity and originality .the 10th edition ( 19021903 ) was just a 10volume supplement to the 25volume 9th edition , but the encyclopedia was rewritten for the 11th edition ( 19101911 ) .we will look at the 9th and 11th editions and how they reflected the confusion of the 19th century .  in the 9th edition, cambridge professor arthur cayley wrote an eleven page article on _ number _ which ` defined ' prime number in a way which appears ( at first ) to omit unity :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a number such as 2 , 3 , 5 , 7 , 11 , & c. , which is not a product of numbers , is said to be a prime number ; and a number which is not prime is said to be composite .a number other than zero is thus either prime or composite . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  after discussing `` complex theories '' ( number rings ) he returns to the ordinary integers and on the same page writes the following .  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ some of these , 1 , 2 , 3 , 5 , 7 , & c. are prime , others , & c. , are composite ; and we have the fundamental theorem that a composite number is expressible , and that in one way only , as a product of prime factors , primes other than 1 ; positive integers ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  he is pressed to this definition , like the table makers , by his decision that integers are all either prime or composite , even unity .lehmer , in the first paragraph of his introduction to his 1914 impressive table of primes , gives exactly this reason for calling unity prime ., however , prefer to exclude unity from the list of primes , thus obtaining a slight simplification in the statement of certain theorems .the same reasons would apply to exclude the number 2 , which is the only even prime , and which appears as an exception in the statement of many theorems also .the number 1 is certainly not composite in the same sense as the number 6 , and if it is ruled out of the list of primes it is necessary to create a particular class for this number alone . '' ] however , in this same edition of the encyclopedia , rev .george marthur s article on _ arithmetic _ , omits unity without apology :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a prime number is a number which no other , except unity , divides without a remainder ; as 2 , 3 , 5 , 7 , 11 , 13 , 17 , & c. [ ] the prime factors of a number are the prime numbers of which it is the continued product .thus , 2 , 3 , 7 are the prime factors of 42 ; 2 , 2 , 3 , 5 , of 60 ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  in the 11th edition , w. f. sheppard s article on _ arithmetic _ excludes unity at first , but then curiously states his definition as unusual !  __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a number ( other than 1 ) which has no factor except itself is called a prime number , or , more briefly , a prime .thus 2 , 3 , 5 , 7 and 11 are primes , for each of these occurs twice only in the table . a number ( other than 1 ) which is not a prime numberis called a composite number . [ ] the number 1 is usually included amongst the primes ; but , if this is done , the last paragraph [ talking about the fundamental theorem of arithmetic ] requires modification ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  g. b. mathews , who wrote the 17page article on _ number _ ( also in the 11th edition ) , defined prime in an ambiguous way with respect to unity , but then continues on , a few lines later , in a way which clearly excludes unity :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a prime number is one which is not exactly divisible by any number except itself and 1 ; all others are composite . [ ] every number may be uniquely expressed as a product of prime factors .hence if is the representation of any number as the product of powers of different primes , the divisors of are the terms of the product their number is and their sum is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  so even as mathematicians were slowly unifying around the modern view , the encyclopdia britannica continued to illustrate the previous century s confusion .at this point it is natural to ask who was the last significant mathematician to define one as a prime .( if we leave off any restriction , it is easy to find recent authors still starting the primes with one : expositions such as m. kraitchik ( 1953 ) , a. beiler ( 1964 ) and c. sagan ( 1997 ) ; as well as the 2011 `` handy science answer book '' and andreasen et al .( 2012 ) . )there is a legend that the last significant mathematician to define one to be prime was henri lebesgue in 1899 ( see ) .however , lebesgue published his first six papers in 18981900 and none of these contain any references to the prime numbers . on the other hand , well after that time g. h. hardy listed one as a prime in several editions of his text _ a course of pure mathematics_. for example , in the first six editions ( 1908 , 1914 , 1921 , 1925 , 1928 and 1933 ) , hardy presented euclid s proof that there are infinitely many primes with a sequence of primes beginning with 1 ( see figure [ fig_hardy6 ] ) .    in the seventh edition ( 1938 ) , this proof is rewritten so that the primes begin at 2 ( see figure [ fig_hardy7 ] ) .( he does the same in a 1929 article ( ) . )hardy ( and his editors ) missed another reference to the unity as prime , he gives the following example of an irrational number : `` the decimal , in which the figure is if is prime , and zero otherwise '' .this example remained the same in all ten editions plus the recent `` revised 10th edition '' printed in 2008 ) .there is no question that for the 17th through early 20th century many mathematicians listed the number one as a prime , but it is also clear this definition was never the unified view of mathematicians .euclid , mersenne , euler , gauss , dirichlet , lucas and landau all omitted one from the primes , so sequence http://oeis.org/a008578 [ ] of , `` prime numbers at the beginning of the 20th century ( today 1 is no longer regarded as a prime ) , '' appears misnamed .we have seen , though , an obvious choice of a new name for this sequence .those with the strongest tradition and argument for including unity were the table makers , because what they were really interested in was factorizations , and hence , the numbers that were not composite .for example , a reprint of brancker and pell s 1688 table ( see figure [ fig_pell ] ) shows that it is a table of incomposites . why not call this sequence the incomposites ?    when we decided to study the development of the definition of prime ( as it applied to finding the smallest prime ) , we did not expect to find such a deep and rich history .it seems so obvious now that one is an integer and that two is the first prime , but we use the concepts of integer and prime with an ease of abstraction that took a millennium to develop .we would like to thank david broadhurst and wilfrid keller for their correspondence on these topics .a. g. aargn and c. r. fletcher , the fundamental theorem of arithmetic dissected , _ math .gazette _ * 81 * ( 1997 ) , 5357 . a. g. aargn and e. m. zkan , a historical survey of the fundamental theorem of arithmetic , _ _ historia math ._ _ * * 28 * * ( 2001 ) , 207214 .j. b. andreasen , l. spalding , and e. ortiz , _ ftce : elementary education k-6 _ , cliffs notes , john wiley & sons , 2011 .barlow , _ an elementary investigation of the theory of numbers , with its application to the indeterminate and diophantine analysis , the analytical and geometrical division of the circle , and several other curious algebraical and arithmetical problems _ , j. johnson and co. , london , 1811 .a. beiler , _ recreations in the theory of numbers _ , dover , 1964 . c. caldwell , a. reddick , and y. xiong , the history of the primality of one a selection of sources , http://primes.utm.edu/notes/one.pdf . ,_ the handy science answer book_ , visible ink press , 2011 .p. a. cataldi , _trattato de numeri perfetti _ , heredi di giovanni rossi , bologna , 1603 .a. cayley , number , in _ the encyclopdia britannica _ , vol .17 , the henry g. allen company , 9th ed . , 1890 ,. 614624 . g. chrystal , _ algebra , an elementary text - book _ , part i , 5th ed . , adam and charles black , london , 1904 .r. crandall and c. pomerance , _ prime numbers : a computational perspective _ , 2nd ed . , springer , 2005 .j. derbyshire , _ prime obsession : bernhard riemann and the greatest unsolved problem in mathematics _ , 2nd ed . ,joseph henry press , 2003 .g. l. dirichlet , _ vorlesungen ber zahlentheorie _ , friedrich vieweg und sohn , braunschweig , 1863 .l. euler , _ elements of algebra _ , 2nd ed ., vol . 1 ,j. johnson and company , 1810 , translated from the french ; with additions of la grange and the notes of the french translator .a. felkel , _ tafel aller einfachen factoren der durch 2 , 3 , 5 nicht theilbaren zahlen von 1 bis 10 000 000 _ , vol . 1, von ghelen , wien , 1776 .r. fricke and f. klein , _ vorlesungen ber die theorie der automorphen functionen _ , vol . 1 , b. g. teubner , leipzig , 1897 . j. glaisher , _ factor table for the fourth million _ , taylor and francis , london , 1879 .c. goldbach , _ letter to euler _ , collected by p. h. fuss in his correspondance mathmatique et physique de quelques clbres gomtres du xviiime sicle , 7 june 1742 . j. p. gram , _ rapport sur quelques calculs 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Abstract: what is the first prime ? it seems that the number two should be the obvious answer , and today it is , but it was not always so . there were times when and mathematicians for whom the numbers one and three were acceptable answers . to find the first prime , we must also know what the first positive integer is . surprisingly , with the definitions used at various times throughout history , one was often not the first positive integer ( some started with two , and a few with three ) . in this article , we survey the history of the primality of one , from the ancient greeks to modern times . we will discuss some of the reasons definitions changed , and provide several examples . we will also discuss the last significant mathematicians to list the number one as prime .


the aim of the virtual observatory ( vo ) is to provide an environment to produce more accurate science .not only does the vo provide an opportunity for technical developments , it also offers unique possibilities for scientific research : the vo framework requires an explicit description of the data that it provides with no implicit assumptions about the data themselves .this requirement is fundamental since interoperability is performed by machine to machine communications without supervision by human beings .such a description is performed by the data model working group in the international virtual observatory alliance ( ivoa ) .the rationale of the data model working group is to provide an abstract description of concepts and their interrelationships , used to fix both the names and meanings of concepts in the vo context and also their internal structure and cross - connections ( ivoa spectral data model 1.01 , 2007)[multiblock footnote omitted ] , that is to describe all the possible universes of classes of data .note that such description and interrelationships are fundamental in scientific research also , with the only difference that , in the last case , researchers are focused on _ particular _ types of objects .figure [ fig : ivoa - sdm ] shows the spectrum data model generated by the data model working group .this data model aims to describe all the universe of tables in the vo that contain a spectrum , hence the data model must not only be valid for observed data , but also for theoretical spectra .this data model ( the generic description of a spectrum ) allows to formulate new questions for researchers that either produce or use theoretical spectra : does this data model describe theoretical spectra ? can all the `` boxes '' in the data model be filled ? and , if this is the case , what is the accuracy of a theoretical model ?do theoretical models have systematic errors ? and , do they have statistical errors ?in addition to the spectrum data model , this working group has also proposed a data model for astrophysical dataset characterization ( ivoa data model for astronomical dataset characterization 1.11 , 2007)[multiblock footnote omitted ] .this data model defines the information required to describe the parameter space of observed or simulated astronomical datasets .[ fig : ivoa - cdm ] is a draft illustration of the description of an atmosphere library like those used in population synthesis .again , a correct characterization of models is fundamental for their application to observations .all theoretical models have a limited range of validity , given by the computational algorithms , the resolution of the input data and other factors .usually , these limitations are described qualitatively in the reference papers , but unfortunately , it is not so common for these limitations to be introduced in the code that computes the models ( fortunately , there are some significant exceptions , like cloudy by * ? ? ? * ) .we stress again that an answer of the previous issues is fundamental in scientific research , since the basic way to obtain the physical parameters of an observed system is to compare observational data with theoretical models .only if these questions are solved in a quantitative way , will it be possible to evaluate the accuracy ( not the precision ) of our knowledge of the observed system .  in the case of stellar populations ,theoretical models are provided by stellar population synthesis .the general case of population synthesis aims to decompose the observed light of the system under study in different stellar populations components , that is the contribution of classes of stars , each class with different effective temperature and gravity . in this case, no further information about the observed system can be obtained unless additional a priori information is considered . in the case of _ evolutionary _ population synthesis , the contribution of the different stellar classes is defined by the evolutionary status of the system , hence the knowledge of the contribution of the different stellar classes can be transformed into knowledge about the age , metallicity , stellar birth - rate ( i.e. star formation history , initial mass function and the amount of formed stars ) of the system .although evolutionary synthesis models are powerful , it is necessary to take into account that :  * the transformation between the contribution of stellar classes and the associated physical parameters requires some a priori hypotheses : a set of atmosphere models , evolutionary tracks and defined star formation properties ( i.e. stellar birth rate ) .* there are different evolutionary conditions that result in the same integrated spectra , so the physical parameters obtained from the analysis of observations are not necessarily unique .these a priori hypotheses , together with the computational algorithms and approximations used by different synthesis codes , define the range of use ( i.e. characterize ) of the results .they are also the principal source of systematic errors in model computations . on the other hand, the degeneracy of the possible solutions produces an intrinsic precision loss in any fit of observational data and theoretical models : the best - fit solutions are not necessarily the correct ones and the distribution of the goodness of fit is also needed for a correct interpretation of the observed data . in order to analyze these issues in detail ,we need to understand what a synthesis model is .the first step needed to evaluate systematic errors , dispersion and the range of application in population synthesis models is to understand how synthesis models work . the main result of synthesis models is the luminosity due to an ensemble of individual stars , each with a given luminosity , .the total luminosity of the ensemble is :    where is the _ estimation of the mean _ luminosity of an individual star .however , in a system where individual stars are not resolved , we can not perform such a sum , since we do not know the individual values . in this case, we can obtain the total luminosity as the composition of different stellar populations ( or different stellar types ) .if we assume that a given stellar type _ can be represented _ by a luminosity and we have some rules that give us the proportion between different stellar types for different physical parameters ( $ ] ) , then    where is the _mean _ ( or average ) luminosity of an average star that represents the ensemble .note that this mean luminosity is not necessarily the luminosity of any _ real _ star , but it is a description of an underlying probability distribution of the possibles luminosities of real stars in the ensemble . as a trivial example , remember the half - alive half - dead schrdinger s cat solution in quantum mechanics .although eq .[ eq : ltot ] and eq .[ eq : lssp ] look similar , they do not provide the same quantity : the first case is the _ actual _ integrated luminosity of a given cluster and the second case is the _integrated luminosity of all the possible clusters with stars and in the same evolutionary conditions .note also that the problem in population synthesis is not the classical statistical one of estimating , but the inverse one : obtaining the _ possible _ theoretical values compatible with the observed integrated luminosity .so , it is necessary to use the complete distribution of theoretical integrated luminosities rather than its mean value .this distribution provides a _metric for goodness - of - fit _ to stellar population studies .   transformed into stars , with a given metallicity and a given age . ]   transformed into stars , with a given metallicity and a wide range of ages . ]  as an illustration of the effects of this intrinsic dispersion , we show in fig .[ fig : sspmean ] the 90% confidence interval of the integrated luminosity of a stellar cluster with transformed into stars for a given stellar birth rate , metallicity and age .it represents the integrated spectrum of _ all _ possible clusters with this number of stars , and evolutionary conditions .[ fig : sspvariance ] show the mean spectrum of a cluster with similar conditions but different ages .the 90% confidence interval in fig .[ fig : sspmean ] , with a single age , covers all the region defined by the mean values of the spectra at different ages .any fit of a real spectrum of such a cluster would produce a very precise result for the age , but the accuracy of this fit can not be known if the underlying probability distributions of integrated luminosities at different ages are not taken into account .we refer to for a more extensive analysis of the probabilistic formulation of population synthesis and for a more complete description about the metrics for goodness of fit .once the origin of the intrinsic dispersion of population synthesis models has been described , we are going to discuss the sources of systematic errors and how to take advantage of the ivoa characterization data model to define the region of application of synthesis models in its parameter space .we refer to for a more complete review on the systematic errors .  in the following, we will only consider the systematic errors related to stellar population synthesis computations itself , assuming that the systematic errors of the inputs are known .the main source of systematic errors are :  1 .atmosphere libraries : systematic errors in atmosphere libraries mainly depend on how realistic our assumption about the classification of stars in the different stellar types are ( see garca - vargas et al .contribution in these proceedings ) .such an evaluation must be implemented by synthesis models makers .+ in addition to the systematic errors , the range of application of synthesis models depends in the coverage on the parameter space ( , , metallicity , ... ) , that is the characterization of the library in this parameter space .tracks / isochrones and their combination with the star formation history : in the case of synthesis models , the shape of isochrones and the _ density _ of stars along the isochrone must fit observational data .small variations in these densities can produce big differences in the models .hence , it is crucial to know the uncertainties in the evolutionary tracks / isochrones used to evaluate the systematic error in synthesis models . currently , there are different groups working on the evaluation of systematic errors in evolutionary tracks , like deglinnocenti et al . and bressan et al .+ additionally , since synthesis models describe the whole population , they also need to take into account evolutionary phases that are included in standard evolutionary tracks , like pre - main sequence evolution , final evolutionary stages like sn or cooling white dwarfs , etc . in general , synthesis modelsmakers combine tracks of different groups to obtain a good coverage in evolutionary phases , but in doing so , they include a potential source of error due to different physics used in track computations. the evaluation of the associated systematic errors and the coverage of the models results due to these effects must be implemented by synthesis models makers .synthesis models algorithms : the final issue is the algorithms used in the model . the main problem is how to combine the different coverage and sampling ( support and precision respectively , following the characterization data model ) of the atmosphere libraries and isochrones in the plane ( see fig . [fig : ivoa - cdm ] ) . in some cases , it is necessary to extrapolate the behavior of atmosphere libraries to regions not covered by the libraries .in addition , there are also differences in how the correspondence of the points in the isochrones and in the atmosphere libraries is done , ranging from how the libraries are interpolated to how the closest model to the given isochrone point is chosen .although the characterization of synthesis models is a task under development , it is currently possible to define its range of application by comparing the coverage of isochrones and atmosphere libraries and using the ivoa characterization data model for this task. it will not be the final solution of the problem of coverage , but it will enormously help towards the use of synthesis models in their correct range of application .synthesis models , as any other theoretical model , are affected by systematic errors .although the evaluation of such errors is a difficult task that only can be performed by synthesis models makers , the current ivoa description of spectrum data model allows to add these errors to the output .  in the case of statistical errors , the spectral data model does not apply directly to theoretical results since the goal of statistics is inferring the underlying distribution from which the data have been drawn , which is assumed to be known in the computation of theoretical models .however , the _ staterror _ box in the current spectral data model can be used to describe the _ intrinsic dispersion _ of theoretical models .the dispersion of synthesis models , although not recognized as such , is computed by several codes under the label of surface brightness fluctuations , so this piece of information is already available .finally , a proper characterization of isochrones and atmosphere libraries coverage allows to obtain a realistic characterization of synthesis models , and its range of application .this research has made use of the spanish virtual observatory supported by the spanish mcyt through grants aya2005 - 04286 and aya2005 - 24102-e .it has also been supported by the spanish mcyt through the project aya2004 - 02703 .mc is supported by a _ramn y cajal _ fellowship .

Abstract: in this contribution we investigate how to describe the results and usage of evolutionary synthesis models . in particular , we look for an explicit and quantitative description of the parameter space of synthesis models and the evaluation of their associated uncertainties and dispersion . first , we need to understand what synthesis models actually compute : we show that a synthetic stellar population with fixed physical parameters ( age , metallicity , star formation history , initial mass function and _ size _ of the system ) can only be described in terms of probability distributions ( i.e. there is an intrinsic dispersion in any model ) . second , we need to identify and characterize the coverage in the parameter space of the models ( i.e. the combinations of input parameters that yield meaningful models ) and the different sources of systematic errors . third , we need a way to describe quantitatively the intrinsic dispersion , the systematic error and the parameter space coverage of the models .  up to now , the parameter space coverage and uncertainties have been described qualitatively in the models reference papers , with potential misinterpretations by models users . we show how virtual observatory developments enable a correct use of synthesis models to obtain accurate ( and not simply precise ) results .


high energy density experiments conducted on large laser facilities are often highly complex in every way .the physics of the target s interaction with the laser , its evolution with time , the relevant material properties at the conditions of interest , and the relation between target properties and observable quantites all require significant effort to describe , and all involve approximations .this is particularly true of so - called ` integrated ' experiments , such as the inertial confinement fusion ( icf ) effort underway at lawrence livermore national laboratory , in which extremely complex targets probe all aspects of the icf approach to nuclear fusion .  in general the only way that such complex systems can be accurately understoodis through computer simulation .these simulations take a range of inputs desribing the target , laser system , and input physics and provide a description of the target evolution .this can then be mapped onto the output of experimental diagnostics .the approach represents a nonlinear , ` black box ' transformation from the uncertain input physics to the measured quantites ; the inversion of this to provide information about difficult physics can be a very difficult statistical problem . in this paperwe discuss the application of bayesian statistics to the solution of this problem , and in particular to the inference of interesting physical parameters from an icf experiment performed at the national ignition facility ( nif ) .the bayesian approach is a well developed statistical method that can be formulated to give powerful results for the inference of interesting quantities , or alternately to provide a rigorous framework for the design of new experimental methods .bayesian theory includes the prior knowledge of the system explicitly and so allows the evolution of interesting quantities to be charted as data accumulates from various sources .the numerical application of the theory is usually through markov chain monte carlo techniques ; in the below we develop a method of incorporating the advantages of the bayesian theory into genetic algorithm approaches which can be parrallelised very easily .these numerical methods have been applied in almost all areas of science for data analysis , including fusion studies , high energy density physics experiments , and hydrodynamic simulations .the example chosen here includes aspects of all of these types of experiment .  in the following section the bayesian approachis set out .particular attention is paid to the interpretation of bayes theorem as an information processing rule , since this can be very useful in understanding the evolution of physical models .the relationship between bayesian methods and the common methodology is discussed , which allows a generalised to be defined that takes into account the effect of nuisance parameters in an experiment , and of prior knowledge of the system .this is applied to a nif ` convergent ablator ' experiment , and used to show the variation in the expected carbon opacity as these bayesian factors are introduced .finally , the mcmc and ga approaches are applied to the same analysis and discussed .in the bayesian approach real numbers are used to describe the level of belief that a statement is true .a set of intuitive axioms govern these numbers , which coincide with the axioms of the usual probability theory . as such, bayesian statistics can be developed as a re - intepretation of frequentist statistics ; rather than the usual understanding that the probability , becomes the belief that the statement is true .this belief depends on some background information , a dependance that is expressed using the conditional probability .an experiment has the effect of updating the background information with some observation , so that after the experiment the belief becomes .the analysis of this updated probability forms the basis of bayesian analysis .the axioms of the theory allow the important bayes theorem to be derived , which relates the probability after the observation of ( the _ posterior _ ) to the probability before ( the _ prior _ ) through the probability of observing the data when the value of is assumed , ( the _ likelihood _ ) .the terms in equation can be viewed as functions of any of their arguments ; the study of as a function of allows the value of to be inferred from data , whereas treating as a function of allows the significance of particular observations to be explored .the latter of these allows the design of experiments by ensuring that an experiment collects the most significant data possible .bayes theorem can be viewed as rule for describing the flow of information in an experiment .the _ information _ associated with an observation of the random variable is where the above is measured in ` nats ' .the more usual unit of ` bits ' is found using a base 2 logarithm .the information associated with an unlikely observation is greater than that with an unsurprising result .bayes theorem becomes and so so the effect of an experiment is to add the new and prior informations .the information defined in equation is evaluated for a given observation result ; when this is not known ( as in problems of experimental design ) it is necessary to consider the expectation value of the information , the _ information entropy _ = -\int p(z){\rm ln } p(z ) dz     \label{eq : information_entropy }     ~{\rm , } \ ] ] which is now a property of the distribution only .cross entropy _ between two distributions and , = -\int p(x){\rm ln } q(x ) dx     \label{eq : information_cross_entropy }     ~{\rm , } \ ] ] measures the common information in the two distributions .this is particularly useful since it is conserved in bayes theorem . using the above definition equationcan be written as = h\left [ p(x\vert d,\mathbb{i } ) \vert p(d\vert x,\mathbb{i } ) \right ]         \notag     ~{\rm , } \ ] ] or = h\left [ p(x\vert d,\mathbb{i } ) \vert p(d\vert x,\mathbb{i})\right ]         \label{eq : kl_distance }     ~{\rm , } \ ] ] so that the increase in cross entropy between the posterior and prior ( the _ kullback - leibler distance _ ) is equal to the cross entropy between posterior and likelihood .experimental design problems often maximise to maximise the effectiveness of an experiment . using the above definitions it has been shown that bayes theorem is the only information processing rule that exactly conserves information cross entropy .the above definitions are all useful forms of the same rule ; the information that a given experiment provides is described by the likelihood function .data analysis is then the analysis of the likelihood function for a given experiment and model , and standard models can be understood as special cases of the likelihood and prior .for example , it is common to use a normally distributed likelihood where is a model for the data given the parameter ; in this case the information associated with an observation is the usual function .the minimisation method is then equivalent to finding the value of the parameter that minimises the information provided by the experiment ( that is , the parameter for which the observation is the least surprising ) . for a constant prior probability ( often used to describe complete ignorance )this is in turn equivalent to finding the value of that has the largest posterior probability .hence the minimisation method can be seen to contain two approximations ; the neglect of the structure of the posterior , and the neglect of prior information .the prior information plays an important role in bayesian analysis . in generalit contains all information that is available before the experiment ; expert opinion , the results from previous experiments , etc .its inclusion allows otherwise very difficult problems to be tackled , and allows the belief in a model or parameter to evolve over time by using the posterior from one experiment as the prior of another . as such data analysis can include the results from entire campaigns of experiments common in modern science , and these experiments need not have the same design or be performed by the same people .all that is required is an expression for the likelihood function for a given experimental design .in this work we aim to investigate the bayesian analysis of inertial confinement fusion experiments performed at the national ignition facility .the usual analysis procedure is to use radiation - hydrodynamic simulations to provide post - shot analysis ; measured quantities like laser power are used as inputs for simulations , the outputs of which are compared to other measured quantities .this comparison is used to gain information about the physics used by simulations , and other unmeasured properties of the experiment .we consider nif shot n110625 , a ` convergent ablator ' experiment designed to mimick the implosion dynamics of a full ignition shot .this shot is of interest since large - scale post shot simulations have been performed in the past , in which a large number of target parameters were varied .this dataset will serve to benchmark the smaller set of simulations used in this study .the eventual aim is that advanced numerical methods such as markov - chain monte carlo ( mcmc ) routines , or genetic algorithms ( ga ) , are used to perform efficient analysis of nif data . to acheive this a robust platform for running hydrodynamic simulationsis required , and the feasibility of such large - scale simulations must be investigated . to that end a set of fortran wrapper routines have been developed that allow the radiation - hydrodynamic code hydra to run as a subroutine to a generic mcmc or ga code .these routines take a set of values of various input parameters ( opacity , equation of state and drive modifiers , etc . ) and return various measurable characteristics of the implosion .modification of input physics databases , and extraction of implosion dynamics , is performed using existing scripts to ensure that returned values are consistent with those commonly used in nif data analysis .calculations are done in parallel , allowing an analysis code to consider several tens of thousands of hydra simulations in one job allocation .  in this worka grid of 1024 simulations of a convergent ablator implosion has been generated , in which a constant multiplier is applied to the absorption opacity of carbon and to the x - ray drive .a representation of the results is shown in figure [ fig : me_vs_dc ] , in which the velocity of the centre of mass of the fuel is plotted as a function of carbon opacity multiplier ( averaging over all other parameters ) .black points show the results of the earlier parameter scan , with the blue line and shading representing a moving average of the mean and standard deviation of those results .the mean and standard deviation of the results of this work are shown in red .the solid line represents the nominal ( unmodified drive ) case , and the broken line shows the drive - averaged case . the large difference between these lines demonstrates the difficulty in increasing implosion velocity in these targets . finally , the black region shows the experimental value .the data in figure [ fig : me_vs_dc ] agree well for the nominal case .the large number of parameters in the original data results in statistical noise in these results , the non - overlapping set of modifications made to simulations ( original simulations focused more on changes to the capsule design than input physics ) , and the differences between the sampling of parameter space mean that the averages do not agree .the good agreement between the current work s nominal case and the previous results demonstrates that the hydra simulations used here are reasonable .it should be noted , however , that the simple modifications to the drive used here do not reflect the current understanding of the x - ray drive in icf experiments , rather they were chosen as the drive multiplier is likely to correlate with the opacity multiplier .we will use three measured results in the calculation of the likelihood ; the implosion velocity when the centre of mass of the fuel reaches a radius of 310 m , the time at which this radius is reached , and the fraction of ablator mass still remaining at this time .experimental data for these are available , and they can be easily extracted from hydra simulations .there are no simulated points that are able to match all three experimental values , making a statistical approach essential .as discussed in the previous section , a simple model for the likelihood is an uncorrelated multivariate normal distribution .the information associated with the likelihood in this case , as a function of the two parameters we wish to infer , is plotted in figure [ fig : normal_likelihood_info ] .the most likely values of the drive and opacity multipliers occur at the minimum point of this surface , when the data are the least surprising .if each of the 3 experimental points are considered individually , each one defines a range of values of opacity and drive multipliers that best match the experimental result .the minimum information point in figure [ fig : normal_likelihood_info ] results from the superposition of the three data points , giving rise to a broad range of ( approximately ) equally likely values of the simulation input parameters .the assumption of a normal distribution for the calculation of the likelihood is based on assumptions about the nature of the _ experimental _ error .simulations are treated as deterministic . in practicethis is not the case ; the values of input parameters will have a probability distribution as well ( reflecting the accuracy with which they are calculated , for example ) and the resulting distribution of code outputs will change the shape of the likelihood function .proper inclusion of all sources of error rapidly increase the parameter space for simulations and so it is advantageous to use an approximate method for some parameters . for thiswe use a normal linear model , which we develop below .  consider the likelihood function for a simulation with two inputs as before , but where one is a ` nuisance ' parameter .if the nuisance parameter is normally distributed , and the response of the simulation to variations in the parameter is linear , then the likelihood function is a normal distribution with a modified width ( see below ) .this linear response method can be used to include the effect of a large number of nuisance parameters with a greatly reduced number of simulations .the potential accuracy of this approach is improved by the fact that input physics models , target fabrication , etc .are highly developed for icf applications making the expected variations in nuisance parameter values small .  to develop the linear response model ,consider a normal approximation in which the likelihood is given by the multivariate distribution with covariance matrix , splitting the simulation input vector into the set of interesting parameters and nuisance parameters , introducing a second multivariate normal distribution ( of covariance ) to describe the variations in nuisance parameters , and treating the response of a simulation as linear , the likelihood function can be found analytically . using ,the result is ( d-\bar{d}(x^{\prime}))}}{\sqrt{(2\pi)^{n_{d}}|\lambda_{d}||\lambda_{y}||\alpha^{t}\alpha| } }      ~{\rm , }     \label{eq : likelihood_marginal}\ ] ] where is the nominal simulation result and the matrices and satisfy the equations in this approximation the likelihood is a normal distribution with a modified covariance matrix .the modification is not diagonal , and so the effect of the simulation response is to distort and rotate the likelihood ; the rotation is towards an axis determined by the strength of the simulation response to the various nuisance parameters and by the expected variation in those parameters . in the case of current interest ,in which experimental data have already been observed , the information associated with the observation ( equation ) is modified to become which defines a modified function that takes into account prior information and the effect of nuisance parameters .these considerations have an important effect on the information associated with a given measurement , and therefore the results of any analysis .the effect of nuisance parameters on n110625 can be included in the above way .we use previous design calculations to populate the response matrix , treating variations in capsule dimensions and chemical composition as nuisance parameters .these are allowed to vary according to their specified manufacturing tolerances .the modification to the likelihood by these parameters is quite significant ; the power of the linear response model is that once the modifications ( in the form of the new covariance matrix ) are known , subsequent analyses do not need to explicitly include the effect of nuisance parameters at all .given knowledge of the prior , from the work in the previous sections and the results of hydra simulations , we can infer the drive and carbon opacity multipliers from the described data , including prior information and the effect of nuisance parameters .the results are shown in figure [ fig : final_info ] ; ( a ) shows the information ( equivalently , the logarithm of the posterior ) where variations in capsule dimensions and atomic composition are neglected , and ( b ) show the results when they are included .both plots use an uncorrelated normal distribution for the prior , with a standard deviation of 0.1 in both directions ; as such both represent a more advanced analysis than the approach shown in figure [ fig : normal_likelihood_info ] .the distortion of the likelihood by nuisance parameters has the effect that the peak in the information is more localised , however the overall distribution is broader suggesting a larger error bar on the inferred values .this can be made more clear by re - casting the data as the posterior probability of the value of the carbon opacity multiplier , by integrating over the drive multiplier .this is not equivalent to the treatment of the drive multiplier as ` nuisance ' ; the detailed response of the system to changes in drive has been included ( not the linear response ) .this approach will prove more accurate for some rapidly varying parameters , however represents a calculation time penalty .the advantage is in the inspection of correlations between parameters that the multi - dimensional analysis allows .the results of integrating over variations in x ray drive are shown in figure [ fig : posterior_c_opac ] , with the blue line showing the inference when nuisance parameters are included , and purple showing the results when they are neglected .there is a significant change in the posterior probability with a corresponding change in the inferred carbon opacity modifier .table [ tab : inferred_c_opacity ] shows the results of this inference for the different bayesian models we have discussed .these results are ( weakly ) sensitive to the choice of prior ; in cases where the priors are taken as centred on 1 , the inferred carbon opacity multiplier is never larger than ..the results of the inference of the carbon multiplier from measured values of the implosion velocity , radius and fraction of ablator remaining in nif convergent ablator shot n110625 .error bars are 68% confidence limits .inference is based on a database of 1024 post - shot hydra simulations , using the bayesian inference models described in the text .the inclusion of prior information and of the variation in capsule dimensions and chemical composition is shown to signficantly reduce the measured carbon opacity multiplier , and the error bar on the result . [ cols="^,^",options="header " , ]   analysis is used ( blue ) .variations in target properties cause a shift in the maximum likelihood estimate of the opacity multiplier , and a change in the shape of the distribution of values . ]the complexity of icf experiments , and of the hydrodynamic simulations used to analyse and design them , make it unfeasible to perform the simple grid parameter scans that have concerned us until now .the statistical framework does , however , allow more advance monte - carlo approaches to be applied , and these have seen great success in the analysis of other large experiments .the application to icf experiments will still require careful consideration , however , and in the following we will discuss the possible approaches that are available .the linear response model described above allows the inclusion of nuisance parameters as described above , however the dependance on parameters that are of experimental interest should be treated more accurately .there is an advantage to keeping as many simulation parameters as possible out of the linear response model ; as we have seen in the previous low dimensional analysis , inspection of the posterior as a function of the model parameters can provide valuable insight .it is not possible to include all parameters however , and so the linear response model will be essential in the implementation of the following advanced methods .the mcmc approach is a statstical method of generating a set of samples of parameter space such that the distribution of samples reflects a given , unknown , probability distribution .the method has been used extensively to probe posterior distributions with very large numbers of parameters ; the potential pitfall is that using hydra simulations to form the step probability in these models , even using the efficient routines applied here , presents a significant computational challenge .the feasability of the mcmc approach can be investigated using a surrogate model , that will approximately reproduce the multi - dimensional behaviour of hydra simulations . in this waythe number of steps required can be found , before resorting to time - consuming hydra simulations .an example solution is shown in figure [ fig : mcmc_posterior ] , where we plot a histogram of mcmc samples from a run of 2000 steps ( initialised at the peak of the prior and with a 500 step ` burn in ' period ) .these simulations use the metropolis - hastings algorithm with a normally distributed jump probability to explore the posterior probability formed from the linear response likelihood and prior described in previous sections .the likelihood is approximated by interpolating in the simulation grid described previously , with very good results .it is well known that even when mcmc simulations poorly describe the multi - dimensional target distribution , low dimensional quantities are often well described .this is seen in our results where the mcmc average and standard deviation carbon opacity multiplier are ; for an mcmc run with 1000 ( post burn in ) samples , this value is which still shows good accuracy .the very small number of simulations required to give a good reproduction of the opacity multiplier is encouraging .it should be noted , however , that even though 2500 hydra simulations are easily acheivable using the parrallel control routines used here , mcmc simulations do not lend themselves to easy parrallelisation .the genetic algorithm approach described in the next section are more ameanable to the large scale simulations necessary here .a ga is a computational tool where a set of samples of parameter space are selected and iterated in order to find the extreme value of some merit function , or fitness .new samples are generated from the old using methods inspired by natural selection - those members that have a large fitness are preferentially selected , and bred together .random jitter is introduced to avoid local maxima in the global fitness , mimiking random mutation in the current population . in data analysis applicationsit is usual to use the inverse function as a fitness , and in this application genetic algorithms are a very robust tool for finding the best fit to data when the parameter space has a high dimension . in previous sections we noted that the usual function can be interpreted as the information in the posterior , and showed how this can be modified to include variations in nuisance parameters and prior information .these considerations allow the application of genetic algorithm techniques to our bayesian analysis .in figure [ fig : ga_meanc ] we plot the evolution of the mean carbon opacity and drive multipliers of a population of 100 parameter space samples as subsequent generations are generated .we use a simple ga , where half the population is carried over between generations and the entire population is allowed to mutate .the breeding population is chosen according to the inverse of the generalised discussed previously .the carbon opacity approaches the expected value of 1.10 quickly , a value that is quite robust as the number of population members ( and therefore hydra calculations ) is reduced . in the caseconsidered here a full ga simulation would require less than 2000 hydra simulations , representing a modest computational problem .the genetic algorithm technique is very easily parrallelised since all members of the sample population are known at the start of each iteration .it has the disadvantage , however , that the converged result gives no information regarding the form of the posterior distribution and so it is difficult to provide confidence intervals for the inferred parameters , or the information entropy in the posterior .a simple exploration of of the space close to the maximum , in order to calculate the hessian matrix , would provide an efficient solution to the problem .we have introduced the bayesian method and developed the normal linear model to describe nuisance parameters in integrated high energy density experiments .this approach allows a modified bayesian function to be defined that efficiently includes prior information and nuisance parameters .this function can be easily applied in any data analysis or experimental design problem , and is particularly well suited to systems with a large number of nuisance parameters .the compact format provided by the modified covariance matrix means that once a sensitivity study has been performed for a given experimental design , the results can be encorporated in all subsequent data analyses in a simple and consistent manner .the expression of the likelihood in these terms also allows powerful bayesian and information - theoretic results to be used .these considerations have been applied to a nif convergent ablator experiment , where it has been shown that they lead to a significant change in the inferred values of the carbon opacity .it has also been shown that the use of computational methods to speed up analysis is very feasible .this capability , along with the bayesian aspects presented here , will allow the integrate analysis of entire experimental campaigns at the nif and to the staistical design of future experiments .the authors would like to thank robbie scott and john castor for help with hydra simulations , and vijay sonnad for stimulating comments .

Abstract: we develop a bayesian inference method that allows the efficient determination of several interesting parameters from complicated high - energy - density experiments performed on the national ignition facility ( nif ) . the model is based on an exploration of phase space using the hydrodynamic code hydra . a linear model is used to describe the effect of nuisance parameters on the analysis , allowing an analytic likelihood to be derived that can be determined from a small number of hydra runs and then used in existing advanced statistical analysis methods . this approach is applied to a recent experiment in order to determine the carbon opacity and x - ray drive ; it is found that the inclusion of prior expert knowledge and fluctuations in capsule dimensions and chemical composition significantly improve the agreement between experiment and theoretical opacity calculations . a parameterisation of hydra results is used to test the application of both markov chain monte carlo ( mcmc ) and genetic algorithm ( ga ) techniques to explore the posterior . these approaches have distinct advantages and we show that both can allow the efficient analysis of high energy density experiments .  inertial confinement fusion , hydrodynamic simulation , bayesian inference , plasma opacity , genetic algorithm , markov chain monte carlo