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Does a combination of proper arithmetic progressions behave Dirichlet-like?

Conjecture: If M is a non-empty finite collection of proper arithmetic progressions, and {x(n)} is the progression created by setting its n-th term equal to the n-th term of a (uniformly) randomly picked member of M, then the probability that {x(n)} contains a prime is unity.

So, my question is, is this in fact true?

Here's my attempt at a proof:

By induction on the cardinality, w, of M:

By Dirichlet's theorem, it is true for w = 1.

Suppose that it is true for w.

Let H be a collection of exactly (w + 1) proper arithmetic progressions.

Case 1. Suppose that one of the members of H has only finitely many terms occurring in {x(n)}. Then, by truncating this initial segment length from consideration, this case collapses to the case where the cardinality is w.

Case 2. Suppose that every member of H has infinitely many terms occurring in {x(n)}. Pick one of the members of H, and block out its terms. The remaining terms constitute a progression satisfying the induction hypothesis.

end of proof

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You refer to $x(n)$ as an arithmetic progression, but there's no reason why it should be an arithmetic progression. –  Gerry Myerson Sep 29 '13 at 0:47
    
@GerryMyerson: You're right. I've made the correction. –  EsperantoSpeaker1 Sep 29 '13 at 12:43
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1 Answer

up vote 2 down vote accepted

Let one of the progressions in $M$ have primes at locations $a_1,a_2,\dots$. The probability that $x(a_1)$ does not come from that progression is $1-(1/m)$, where $m$ is the number of progressions in $M$. The probability that none of the terms $x(a_1),x(a_2),\dots,x(a_r)$ comes from that progression is $(1-(1/m))^r$. This goes to zero as $r$ increases, so the probability that there is a prime in $x(n)$ is $1$.

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