An Introduction to Number Theory (Graduate Texts in by G. Everest, Thomas Ward

By G. Everest, Thomas Ward

Comprises updated fabric on contemporary advancements and subject matters of vital curiosity, similar to elliptic services and the hot primality try out Selects fabric from either the algebraic and analytic disciplines, providing numerous various proofs of a unmarried consequence to demonstrate the differing viewpoints and provides sturdy perception

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As a simple illustrative example, let S denote the set {2} and let ZS denote the ring Z[ 12 ] consisting of all rational numbers with a denominator consisting of a power of 2. Given any nonzero q ∈ Q, write q = 2r q , where r ∈ Z and the numerator and denominator of q are odd. Define the S-norm of q to be |q|S = |q |. The ring R has infinitely many units, consisting of the rational numbers ±2k for k ∈ Z. The ring R is sometimes called the ring of S-integers of Z, and its units are known as S-units.

Now 23n + 1 is a prime √ smaller than 223 − 1 only for n = 2, 12, 20, 26, 30, 36, 42, 44, 50, 56, 60, 62, 72, 84, 86, 102, 104, 110. Trial division shows that M23 is divisible by the first of the resulting numbers, 47. In general, there is no reason to expect the smallest possible candidate to be a divisor, but even if the largest were the first such divisor, only 18 trial divisions are involved. In 1876, Lucas discovered a test for proving the primality of Mersenne numbers. Using this test, he proved that 2127 − 1 = 170141183460469231731687303715884105727 is prime, but 267 − 1 is not.

A) If q = 2p − 1 is a Mersenne prime, prove that 2p−1 q is a perfect number. (b) Prove that if n is an even perfect number, then n has the form 2p−1 (2p −1) for some prime of the form 2p − 1. It is not known if there are any odd perfect numbers, but there are certainly no odd perfect numbers smaller than 10400 . Write Mn = 2n − 1 for the nth Mersenne number. The Mersenne numbers have special properties that make them particularly suitable for primality testing. The next result is the first of a series of results showing that divisors of Mn are quite prescribed when n is prime.

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