Download The Book of Prime Number Records by Paulo Ribenboim (auth.) PDF

By Paulo Ribenboim (auth.)

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Extra resources for The Book of Prime Number Records

Example text

As a matter of fact, this will be shown in Chapter 6, Section II, under the assumption of an unproved hypothesis. And how a bout the valence of the valence function N~? I have already said that there exist infinitely many m that are not values of~, for which N ~(m) = O. So N ~ assumes the value 0 infinitely often. This was generalized by Erdos in 1958: If s ~ 1 is a value of N~, then it is assumed infinitely often. ) In Chapter 4, Section I, I shall return to Euler's function to study its properties of distribution.

If none does, then N is a prime. If, say, No divides N, write N = NONl' so Nl < N, and then repeat the same procedure with No and with Nl' Eventually this gives the complete factorization into prime factors. What I have just said is so evident as to be irrelevant. It should, however, be noted that for large numbers N, it may take a long time with this algorithm to decide whether N is a prime or composite. This touches the most important practical aspect, the need to find an efficient algorithm - one which involves as few operations as possible, and therefore requires less time and costs less money to be performed.

Wilson's theorem, which characterizes prime numbers, might seem very promising. But, it has to be discarded as a practical test, since the computation of factorials is very time consuming. Fermat's little theorem says that if p is a prime and a is any 37 III. Classical Primality Tests Based on Congruences natural number not a multiple of p, then a P- 1 == I (mod p). However, I note right away that a crude converse of this theorem is not true because there exist composite integers N, and a ~ 2, such that a N - 1 == 1 (mod N).

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