Primes in the intersection of arithmetic progressions

Asked Dec 08 '21 at 14:33

Active Dec 08 '21 at 14:33

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Let $a,b,r,s$ be given constants. We know that that the arithmetic progressions $\{ax + r : x \in \Bbb Z\}$ and $\{by + s : y \in \Bbb Z\}$ intersect if and only if $\gcd(a, b) \mid (s − r)$.

In this case, I am asking if this intersection contain at least one prime number. I know about the Dirichlet theorem, but I am not able to apply it here.

asked Dec 08 '21 at 14:33

Safwane

3,840

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The intersection is $m\mathbb{Z}+t$ where $t$ is an integer congruent to $r$ mod $a$, $s$ mod $b$, and $m$ is the lcm of $a,b$. What’s your obstacle to applying Dirichlet? – Aphelli Dec 08 '21 at 14:36
@Mindlack: Can you convert this into an answer. – Safwane Dec 08 '21 at 14:40
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The question "before" this is this one. – Dietrich Burde Dec 08 '21 at 14:40
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$4\mathbb Z$ and $6\mathbb Z$ intersect, but the intersection does not contain primes. – markvs Dec 08 '21 at 14:44

Primes in the intersection of arithmetic progressions

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