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.
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 $a$?