I have been struggling with this question. I don’t know how to proceed. Would you be able to walk me through the steps to complete it?
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1it seems to be false, $(p+1)^2$ is always $0$ or $1\bmod 3$ – Asinomás Apr 20 '20 at 17:09
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1$(p^2)+2p \equiv-1 \bmod 6$ iff $(p+1)^2 \equiv 0 \bmod 6$ iff $p \equiv -1 \bmod 6$. See https://math.stackexchange.com/a/671854/589 and https://math.stackexchange.com/questions/407259/how-do-you-prove-that-there-are-infinitely-many-primes-of-the-form-5-6n – lhf Apr 20 '20 at 17:26
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There are infinitely many primes of the form $p=6k+5$. – rtybase Apr 20 '20 at 17:28