Problem: Prove that there are infinitely many primes of the type 5 mod 6.
My professor did the problem and the proof was horribly long. Can someone show me a shorter version of the proof of this problem.
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$\begingroup$Hint: If $n\equiv 5\pmod 6$, then for some prime factor $p$, $p\equiv 5\pmod 6$.
Then try a variation of Euclid's proof that there are infinitely many primes.
It is harder to prove that there are infinitely many primes $\equiv 1\pmod 6$.
$\endgroup$ $\begingroup$If we cross out from sequence of positive integers all numbers divisible by 2 and all numbers divisible by 3 then all remaining numbers will be in one of two forms:
$S1(n)=6n+5=5,11,17,...$ or $S2(n)=6n+7=7,13,19,...;$$n=0,1,2,3,...$
So all prime numbers also will be in one of these two forms.
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