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Unformatted text preview: Mersenne primes and Even Perfect numbers Jonathan L.F. King University of Florida, Gainesville FL 326112082, 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 nontrivially 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.Coprimeness Lemma....
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This note was uploaded on 01/26/2012 for the course MHF 3202 taught by Professor Larson during the Fall '09 term at University of Florida.
 Fall '09
 LARSON

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