By Abraham A. Fraenkel
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Sample text
W e must not forget, however, that con clusions formed inductively, even when based on a huge number of experiments, cannot yet be considered as established. In the theory of numbers, their validity is much less certain than in the experi mental sciences. W e m ay accept a conclusion only after it has been proved mathematically. Let us consider a few instances. Euler found, b y testing all the integers up to 2500, that it was apparently possible to express every odd number as the sum o f a prime number, p, and of the double of a square: n = p + 2m2.
8 A sequence like (1, 3,5, . . ) is called an arithmetical progression. In any arithmetical progression the difference between tw o consecutive elements is con stant; in the sequence (1, 3, 5, . . ) it is equal to 2. W e can, therefore, write the progression in the form (1 + 2n) where n denotes any natural number, including 0. The fundamental theorem states, therefore, that in the arithmetical progression (1 + 2n) there are infinitely many primes. W ith this as a starting point it appears reasonable to pose the follow ing question, b y way of generalization: Let a and d be tw o natural numbers.
W ilson’s Theorem is a criterion for prime numbers: a natural number p is prime if (and only if) the above-mentioned sum is divisible b y p. The reader should attempt to prove this b y himself. The last theorems can be more simply formulated if we employ a definition which Gauss placed at the beginning of the theory of num bers. W e write a = b (mod. m) or, in words, a is “ congruent” to b modulo m, if the difference a — b is divisible b y the natural number m. In this case, a and b when divided b y m yield equal remainders.