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单词 MersenneNumbersTwoSmallResultsOn
释义

Mersenne numbers, two small results on


This entry presents two simple results on Mersenne numbers11In this entry, the Mersenne numbers are indexed by the primes., namely that any two Mersenne numbers are relatively prime and that any prime dividing a Mersenne number Mp is greater than p. We prove something slightly stronger for both these results:

Theorem.

If q is a prime such that qMp, then p(q-1).

Proof.

By definition of q, we have 2p1(modq). Since p is prime, this implies that 2 has order p in the multiplicative groupMathworldPlanetmath q{0} and, by Lagrange’s Theorem, it divides the order of this group (http://planetmath.org/Group), which is q-1.∎

Theorem.

If m and n are relatively prime positive integers, then 2m-1 and 2n-1 are also relatively prime.

Proof.

Let d:=gcd(2n-1,2m-1). Since d is odd, 2 is a unit in d and, since 2n1(modd) and 2m1(modd), the order of 2 divides both m and n: it is 1. Thus 21(modd) and d=1.∎

Note that these two facts can be easily converted into proofs of the infinityMathworldPlanetmathPlanetmath of primes: indeed, the first one constructs a prime bigger than any prime p and the second easily implies that, if there were finitely many primes, every Mp (since there would be as many Mersenne numbers as primes) is a prime power, which is clearly false (consider M11=2389).

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更新时间:2025/5/4 11:35:04