Pythagoras, a native of the Greek island of Samos (he lived from about 570 B.C. to 500 B.C.), migrated to Crotona in southern Italy where he founded an academy of learning which brought him a devoted following and lasting fame. All the discoveries of the school were, by custom, attributed to Pythagoras himself. Just what his personal contributions are is almost impossible to estimate. However, in total, the school's achievement was great, practically marking the advent of deductive mathematics. In the hands of the Pythagoreans mathematics was directed along various channels, some of which have not yet dried up. In this section, we consider several Pythagorean topics and some later developments.

In the terminology of the Greeks, “arithmetic” is equivalent to our number theory, while “logistic” was their term for practical calculations.

Amicable Numbers. Two positive integers constitute an amicable pair (friendly pair) if the proper divisors of each one add up to the other. (The proper divisors do not include the number itself.) The smallest pair, the only one known to the Pythagoreans, is (220,284):

220 has divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, whose sum is 284; 284 has divisors 1, 2, 4, 71, 142, whose sum is 220.

The next new pair was announced in 1636 by the celebrated French genius Pierre de Femat (1601–1665); it is (17296, 18416). In 1638 Descartes gave a third pair. In 1747 Euler gave 30 pairs, and in 1750 he increased that number to 60 pairs. Today over 900 pairs are known.