TITLE: Infinitely many primes p that are not congruent to 1 mod 5
QUESTION [5 upvotes]: Argue that there are infinitely many primes p that are not congruent to 1 modulo 5.
I find this confusing.  Is this saying $p_n \not\equiv 1 \pmod{5}$?
To start off I tried some examples.
$3 \not\equiv 1 \pmod{5}$
$5 \not\equiv 1 \pmod{5}$
$7 \not\equiv 1 \pmod{5}$
$11 \equiv 1 \pmod{5}$
$13 \not\equiv 1 \pmod{5}$
$17 \not\equiv 1 \pmod{5}$...
If this is what the question is asking i've come to the conclusion that this is true.   Either way, i've got no clue how to write this as a proof.

REPLY [2 votes]: Hint $\rm\ \ 5n^2\!-n\: $ has a larger set of prime factors $\rm\not\equiv 1\ mod\ 5\:$ than does $\rm\:n.$