By I.G. Macdonald

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6) MiYi == 1 (mod m;). 4), we choose i E {l, ... , k}. 6) by Ti. 4). Then Xl == X2 (mod m;) for each i = 1, ... , k. Since mi are pairwise coprime, we have Xl == X2 (mod M). 4) is uniquely determined modulo M. 8. 3), where gcd(3,5, 7) = 1, we find the solution X = 23, which is the smallest in positive integers. 5 illustrates the cube 23 x 23 x 23, which is decomposed into blocks. Most of them are 3 x 5 x 7 blocks associated with the moduli 3, 5, and 7. There is one 2 x 3 x 2 block associated with the remainders 2, 3, and 2, and other mixed blocks associated with both moduli and remainders.

5. Primality of Fermat Numbers I have found that numbers of the form 22 = + 1 are always prime numbers and have long since signified to analysts the truth of this theorem . Pierre de Fermat in his letter to Father Marin Mersenne on December 25, 1640, [Mahoney, p. 140J. 1. Notice that the number 223 + 1 = 28 + 1 is prime, but the numbers 23 + 1 and 228 + 1 are composite (cf. Appendix A). This example shows that if 2n + 1 is prime, then 2 2n + 1 need not be prime and vice versa (see [Sierpinski, 1970, Problem 141]).

13. 11) = ordda, then 18 17 lectures on Fermat numbers for n = ke, k E {I, 2, ... 11) holds only for these exponents. Proof. If n = ke, an _ 1 then = a ke _ 1 = (a e _ 1) ( a e(k -1) + ... 11) is valid due to the previous definition. Assume, for an instant, that d I a ke + h -1 for some k E {I, 2, ... } and 0 Then (a ke + h _ 1) - (a ke - 1) = ake(a h - 1). < h < e. 11), gcd(d, a ke ) = 1, we get that a h -1 is divisible by d. This contradicts the minimality of e. 0 Let p be a prime. By Fermat's little theorem, the maximum order modulo p of any integer a coprime to p is p - 1.