rch for large primes goes on: Below is a list of the largest found to date. For a great deal of information on primes, including the numbers below, check out the The Primes Home Page A special method, the Lucas-Lehmer test has been derived to check primality.
2^859433-1 (258,716 digits); Slowinski and Gage, 1994 2^756839-1 (227832 digits); Slowinski and Gage, 1992 391581*2^216193-1 (65087 digits); Noll and others, 1989 2^216091-1 (65050 digits); Slowinski, 1985 3*2^157169+1 (47314 digits); Jeffrey Young, 1995 9*2^149143+1 (44898 digits); Jeffrey Young, 1995 9*2^147073+1 (44275 digits); Jeffrey Young, 1995 9*2^145247+1 (43725 digits); Jeffrey Young, 1995 2^132049-1 (39751 digits); Slowinski, 1983 9*2^127003+1 (38233 digits); Jeffrey Young, 1995
Subdivisions of even numbers are reported by Nicomachus (a neo-Pythagorean, ~100 A.D.).
Similar subdivisions of odd numbers are:
Actually the third category is wholy subsumed by the second.
Also ascribed to the Pythagorean s is the study of perfect and amicable and deficient numbers.
A number n is perfect if the sum of its divisors is itself: Examples: ( 6, 28, 496, 8128, ...) In Euclid, we find the proposition: If is prime, then is perfect. (Try , p= 2, 3, 5, 7 to get the numbers above.)
The pair of numbers a and b are called Amicable If the divisors of a sum to b and if the divisors of b sum to a. Example: 220 and 284.
In addition, the number a was classified as abundant or deficient according as their divisors summed greater or less than a, respectively.
Example: 12-divisors are: 6,4,3,2,1-- . So, 12 is abundant.
Example: All primes are deficient.
Just the fact of finding perfect numbers using the previous propositions has spawned a cottage industry of determining those numbers p for which is prime. Such primes are called Mersenne (1588-1648) primes after the friar of the 17 century So far 33 have been found, though it is unknown if there is another between the 32nd and 33rd. It's not known if there are an infinity.
number p year by ---------------------------------------------------------- 1-5 2,3,5,7,13 in or before the middle ages 6-7 17,19 1588 Cataldi 8 31 1750 Euler 9 61 1883 Pervouchine 10 89 1911 Powers 11 107 1914 Powers 12 127 1876 Lucas 13-14 521,607 1952 Robinson 15-17 1279,2203,2281 1952 Lehmer 18 3217 1957 Riesel 19-20 4253,4423 1961 Hurwitz & Selfridge 21-23 9689,9941,11213 1963 Gillies 24 19937 1971 Tuckerman 25 21701 1978 Noll & Nickel 26 23209 1979 Noll 27 44497 1979 Slowinski & Nelson 28 86243 1982 Slowinski 29 110503 1988 Colquitt & Welsh jr. 30 132049 1983 Slowinski 31 216091 1985 Slowinski 32? 756839 1992 Slowinski & Gage 33? 859433 1994 Slowinski and Gage,
Figurate Numbers. Numbers geometrically constructed had a particular importance to the Pythagorean s.
Triangular numbers. These numbers are 1, 3, 6, 10, ... . The general form is the familiar
Square numbers These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots, they prove(?) the well known formula
The gnomon is basically an architect's template that marks off "similar" shapes.
Note the gnomon has been placed so that at each step, the next odd number of dots is placed.
Figurate Numbers of any kind can be calculated.
The Pentagonal and Hexagonal numbers are shown in the graphs below.
Note that the sequences have sums given by
Similarly, polygonal numbers of all orders are dsignated; this process can be extended to three dimensional space, where there results the polyhedral numbers. Philolaus is reported to have said:
All things which can be known have number; for it is not possible that without number anything can be either conceived or known.
The Pythagorean Theorem
Whether Pythagoras learned about the 3, 4, 5 right triangle while he studied in Egypt or not, he was certainly aware of it. This fact though could not but strengthen his conviction that all is number. It would also have led to his attempt to find other forms. How might he have done this?
One place to start would be with the square numbers, and arrange that three consecutive numbers be a P