Today, I came across this question while learning sequence and series, the question is Prove that we cannot make an infinite AP in the set of prime integers. I tried to solve it, but I couldn't proceed as there is no general formula for prime numbers so how can I prove that no infinite AP will be possible, so I resorted to proving that any AP which has all terms prime integers will end at kth term.
I also consulted the Wikipedia article of Green-Tao Theorem (https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem) but I wasn't able to understand the solution present there.
I also saw some StackExchange answers but none answered my question or at least gave me a hint.
I also read this research paper http://academic.csuohio.edu/soprunov_i/pdf/primes.pdf
So I would request you to please guide me with a hint, not a solution.
Please note this is not my homework exercise, I am just learning algebra for my competitive examination preparation.
The set of primes contains arbitrarily long APs by the Green-Tao theorem, but clearly no infinite AP. Assume that $p_0$ and $p_1$ are the first elements of such sequence and let $d=p_1-p_0$. $p_0+p_0 d=p_{p_0}$ is an element of such sequence but cannot be a prime, since it is a multiple of $p_0$.