By G. Everest, Thomas Ward

Contains up to date fabric on fresh advancements and issues of important curiosity, comparable to elliptic capabilities and the recent primality try Selects fabric from either the algebraic and analytic disciplines, proposing numerous diversified proofs of a unmarried consequence to demonstrate the differing viewpoints and provides strong perception

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**Extra resources for An Introduction to Number Theory (Graduate Texts in Mathematics)**

**Sample text**

J ,X 3, with π(1) = 0, π(2) = 1. (b) Deﬁne a function f by f (x, x) = 0 and f (x, y) = 1 2 1+ x−y |x − y| for x = y. 9 to prove that 2n pn = 1 + f (n, π(j)). 19 seems to oﬀer a general primality test because the condition is necessary and suﬃcient. The problem is that in practice it is impossible to compute (n − 1)! modulo n in a reasonable amount of time 34 1 A Brief History of Prime for any integer that is not quite small. In Chapter 12 we will seek to give a better understanding of what counts as “small” or “large” in terms of modern computing.

D) What can you deduce if n has three distinct prime divisors? Zsigmondy’s Theorem holds in greater generality, though we will not prove the following result here. 16. [Zsigmondy] Let an = cn − dn , where c > d are positive coprime integers. Then an always has a primitive divisor unless (1) c = 2, d = 1 and n = 6; or (2) c + d = 2k and n = 2. 17. Find some nontrivial examples of case (2) of the theorem. 19 on p. 169. 18. Prove that the sequence (un ) does not satisfy a Zsigmondy Theorem in each of the following cases.

Fermat’s Little Theorem says that, for any prime p, 2p−1 − 1 is divisible by p. It sometimes happens that 2p−1 −1 is divisible by p2 . Find all the primes p with this property for p < 106 . Such primes are called Wieferich primes, and it is not known if there are inﬁnitely many of them. 12. *A pair of congruences that arises in the Catalan problem (see p. 57) for odd primes p, q is pq−1 ≡ 1 (mod q 2 ) and q p−1 ≡ 1 (mod p2 ). 22) is called a Wieferich pair. Find all the Wieferich pairs with p, q < 104 .