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Mersenne numbers, two small results on∗
CWoo†
2013-03-21 18:55:08
This entry presents two simple results on Mersenne numbers1 , 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 q | Mp , then p | (q − 1).
Proof. By definition of q, we have 2p ≡ 1 (mod q). Since p is prime, this implies
that 2 has order p in the multiplicative group Zq \ {0} and, by Lagrange’s
Theorem, it divides the order of this 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 Zd and, since
2n ≡ 1 (mod d) and 2m ≡ 1 (mod d), the order of 2 divides both m and n: it
is 1. Thus 2 ≡ 1 (mod d) and d = 1.
Note that these two facts can be easily converted into proofs of the infinity
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 = 23 · 89).
∗ hMersenneNumbersTwoSmallResultsOni created: h2013-03-21i by: hCWooi version:
h36874i Privacy setting: h1i hResulti h11A41i
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1 In this entry, the Mersenne numbers are indexed by the primes.
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