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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.


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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

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up vote 4 down vote accepted

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 $a$?

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Thank you, I should have noticed this. A bit too late in the evening to be doing any of this. –  user6323 Jan 28 '11 at 0:00
it is not true that a should be prime. a must be odd. 21 is an odd and not prime. 21+2 = 23 is prime. so 21+2k, k=0,1,2,3... contain primes. –  Ron Ald Mar 11 at 13:24

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