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mersenne-perfect

# mersenne-perfect - Mersenne primes and Even Perfect numbers...

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Mersenne primes and Even Perfect numbers Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 11 October, 2009 (at 16:55 ) Prolegomenon. Here is some notation 1 used in the sequel. Employ P for the set of primes, and D for the set of odd posints. For n a positive integer, use M n := [2 n - 1] to name the n th Mersenne number . 1 : Lemma. If M n is prime then n is prime. Proof. Proving the contrapositive, suppose n factors non-trivially as n = K · L , where K,L [ 2 .. ) . Recall the polynomial identity x L - 1 = [ x - 1] · [1 + x + x 2 + · · · + x L - 1 ] . Setting x := 2 K gives factorization that M n = A · B , where A := 2 K - 1 note > 3, and B = X L - 1 =0 2 ‘K note > 1 + 2 1 · 2 = 5 . Hence M n is composite. 2a : Lemma. For each posint k : σ ( k )= k +1 IFF k P . 2b : Lemma. n,k Z , nec. Gcd( n, k ) •| [ n - k ] . In particular, [ k +1] k . 2c : Coprimeness Lemma. Suppose d,x, y are integers, k N and p P . Then: i : If d •| [ xy ] and d x , then d •| y .

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