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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 statement –  Ethan Dec 17 '12 at 23:22
    
@Ethan, I've added an explanation. –  lhf Dec 17 '12 at 23:23
    
but im multiplying p by 2, i dont understand –  Ethan Dec 17 '12 at 23:24
    
@Ethan, $2p \equiv 1 \bmod 3$ implies $2\cdot 2p \equiv 2 \cdot 1 \bmod 3$. –  lhf Dec 17 '12 at 23:25
    
oh I see, im being an idiot lol –  Ethan Dec 17 '12 at 23:27

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