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Unformatted text preview: A special case of Dirichlet’s theorem Jonathan L.F. King [email protected] 28 October, 2009 (at 15:16 ) Entrance. An arithmetic progression ( A.P. ) means a set T + M Z of integers, ♥ 1 where the gap ( or modulus ) M is a posint and translation ( or target ) T is an integer. I’ll also use comb for “arithmetic progression”. An A.P. C := T + M Z is coprime if T ⊥ M . 1 : Dirichlet’s Theorem. Each coprime arithmetic pro- gression contains infinitely many prime numbers. ♦ While this is difficult to prove in general, there are three easy special ♥ 2 cases, the combs 1 + 3 Z 1 + 4 Z and 1 + 6 Z . We will establish this last case. Henceforth, let ≡ mean ≡ 6 and let “ congruent ” mean “mod-6 congruent”. 2a : Lemma. Suppose a product q 1 · q 2 · ... · q ‘ · ... · q L of integers ♥ 3 is coprime to 6 . Then each multiplicand q ‘ is coprime to 6 . ♦ Proof. Exercise ; prove the contrapositive. Where does your argument use that each q ‘ is is an integer ? Does the lemma generalize to “6” being replaced by N , an arbitrary posint? Henceforth, symbols r and q , with or without ap- pendages, range over the integers....
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