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How does dirichlets theorem apply to the question of weather or not there are an infinite number of primes p, such that $ap\equiv b$ mod c, for some constants a,b,c.

For example consider the question of weather or not there are an infinite number of primes p such that, $2p\equiv 1$ mod 3

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Because $2\cdot 2 \equiv 1 \bmod 3$, your example can be phrased as whether there is an infinite number of primes $p$ such that $p \equiv 2 \bmod 3$. This is the same as considering the primes in the arithmetic progression $3k+2$, to which Dirichlet's theorem applies.

The general case is essentially like this, except that you need to consider $d=\gcd(a,c)$. If $d$ does not divide $b$ then there are no solutions, let alone primes. Otherwise, you can divide by $d$ and then invert $a/d$ modulo $c/d$ to get an arithmetic progression.

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    I dont understand how asking if there are an infinite number of primes p such that 2p is congruent to 1 mod 3 is the same as your statement2012-12-17
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    @Ethan, I've added an explanation.2012-12-17
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    but im multiplying p by 2, i dont understand2012-12-17
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    @Ethan, $2p \equiv 1 \bmod 3$ implies $2\cdot 2p \equiv 2 \cdot 1 \bmod 3$.2012-12-17
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    oh I see, im being an idiot lol2012-12-17
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    Lets say we didn't know if there were an infinite number of primes p such that $p\equiv b$ mod a, would saying there are an infinite number of primes p such that $ap \equiv b$ mod c, imply the former2012-12-17
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    @Ethan, yes, unless there are no primes, as I've mentioned.2012-12-17
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    how would the infinititude of primes p, such that ap is congruent to b mod c, imply the infinitude of primes p such that p is congruent to b mod c2012-12-17
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    @Ethan, I suggest you post a separate question.2012-12-17