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 is greater than . We prove something slightly stronger for both these results:
If is a prime such that , then .
If and are relatively prime positive integers, then and are also relatively prime.
Let . Since is odd, is a unit in and, since and , the order of divides both and : it is . Thus and . ∎
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 and the second easily implies that, if there were finitely many primes, every (since there would be as many Mersenne numbers as primes) is a prime power, which is clearly false (consider ).
|Title||Mersenne numbers, two small results on|
|Date of creation||2013-03-22 15:07:53|
|Last modified on||2013-03-22 15:07:53|
|Last modified by||CWoo (3771)|