e of the aristocrasy. Probably because of this, after a time the citizens turned against him and his followers, burning his house. Forced out, he moved to Metapontum, also in Southern Italy. Here he died at the ages of eighty. His school lived on, alternating between decline and re-emergence, for several hundred years.
Tradition holds that Pythagoras left no written works, but that his ideas were carried on by his eager disciples.
What is known of the Pythagorean school is from a book written by the Pythagorean, Philolaus of Tarentum. From this book Plato learned the philosophy of Pythagoras.
The dictum of the Pythagorean school was All is number.
What this meant was that all things of the universe had a numerical attribute that uniquely described them. For example,
Classification of numbers. The distinction between even and odd numbers certainly dates to Pythagoras. From Philolaus, we learn that
"...number is of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two; and of each kind there are many forms."And these, even and odd, correspond to the usual definitions. But even-odd means a product of an even and odd number. Note: orginally the number 2 was not considered even.
Prime or incomposite numbers and secondary or composite numbers are defined in Philolaus:
Proposition. There are an infinite number of primes.
Proof. (Euclid) Suppose that there exist only finitely many primes . Let . The integer N-1, being a product of primes, has a prime divisor in common with N; so, divides N- (N-1) =1, which is absurd!
The search for primes goes on. Eratsothenes (276 B.C. - 197 B.C.), who worked in Alexandria, devised a seive for determining primes. This seive is based on a simple concept:
Lay off all the numbers, then mark of all the multiples of 2, then 3, then 5, and so on. A prime is determined when a number is not marked out. So, 3 is uncovered after the multiples of two are marked out; 5 is uncovered after the multiples of two and three are marked out.
The Primal Challenge
The search 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.
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