# A Selection of Problems in the Theory of Numbers. Popular - download pdf or read online By Waclaw Sierpinski, I. N. Sneddon, M. Stark

ISBN-10: 0080107346

ISBN-13: 9780080107349

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Example text

2)(,-l). —1)/2]! ]2 on dividing by p gives the same remainder as (p— 1)!. ]2+l is divisible by p. We have thus proved THEOREM 16. ]2+l is divisible by p. In order to deduce a further corollary from this theorem, we prove the following LEMMA. If p is a prime number and a an integer not divisible by p, then there exist natural numbers x and y, x < yfp and y < Jp, such that for a suitable sign + or — the number ax±y is divisible by p. Proof Let p be a given prime number and let m denote the greatest natural number ^s/p; then m + l >sfp so that ( m + l ) 2 > p.

2 1 9 4 7 +1 having 587 digits (Robinson ). However, we do not know any other prime divisors of the number Fl945 or its decomposition into prime factors (see § 22). ,qs form an increasing sequence). The proof of unicity depends on some simple theorems on prime numbers. THEOREM 5. The prime number p has only two natural divisors: 1 and p. b, where b is a natural number > 1, because in case b = 1, p would be equal to a, contrary to the supposition about the number a. The number p would then be the product of two natural numbers 38 PROBLEMS IN THE THEORY OF NUMBERS greater than 1, contrary to the supposition that/?

So, for a certain natural number k, rfk) will be the last term of this sequence; then certainly n{k) = 1, because in case rfk) > 1 we could subdivide n(k) =p

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### A Selection of Problems in the Theory of Numbers. Popular Lectures in Mathematics by Waclaw Sierpinski, I. N. Sneddon, M. Stark

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