# arithmetic progression of primes

Prove that there is no arithmetic progression that consists only of primes.

A question that I've been set; I'm guessing it makes use of primes being written in the form 4k+1 and 4k+3? Not sure where to start.

Thanks.

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Play with some special cases. Can you find an arithmetic progression of primes with common difference 1? Why not? What about 2? 3? Can you find an argument that generalizes to any common difference d? – Qiaochu Yuan Jan 27 '11 at 19:43

An arithmetic progression is a sequence of the form $$a,\quad a+b,\quad a+2b,\quad a+3b,\quad a+4b,\quad\ldots, a+nb,\quad\ldots$$ with $b\neq 0$.
If $a$ is not prime, you're done. So suppose $a$ is a prime. Can you find a term later in the sequence that is guaranteed to be a multiple of $a$?