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Prove that there are infinitely many primes of the form $5k+3$

I known this is Dirichlet's Theorem Special case, but I want to find a similar following elementary proof：

1:there are infinitely many primes of the form $8k+3$, consider $x^2+2\equiv 0\pmod p$

2: there are infinitely many primes of the form $5k+4$, consider $x^2-5\equiv 0\pmod p$

3: there are infinitely many primes of the form $7k+6$, consider $x^3+x^2-2x-1\equiv 0\pmod p$

Martin Sleziak
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math110
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  • What does $0\pmod{p}$ mean here? – barak manos Sep 28 '16 at 07:00
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    Similar questions have come up before, e.g., http://mathoverflow.net/questions/15220/is-there-an-elementary-proof-of-the-infinitude-of-completely-split-primes and http://mathoverflow.net/questions/16735/is-a-non-analytic-proof-of-dirichlets-theorem-on-primes-known-or-possible and http://mathoverflow.net/questions/28160/is-there-another-proof-for-dirichlets-theorem and http://mathoverflow.net/questions/32624/special-cases-of-dirichlets-theorem and https://www.jstor.org/stable/2691037?seq=1#page_scan_tab_contents and https://web.math.pmf.unizg.hr/nastava/studnatj/dirichleteuclid.pdf ... – Gerry Myerson Sep 28 '16 at 07:31
  • ... and basically just type "special cases of Dirichlet's Theorem" into the internet and see what comes back at you (and then write up a brief summary of what you've learned, and post it as an answer). – Gerry Myerson Sep 28 '16 at 07:32
  • Thanks,@GerryMyerson,I will learned. – math110 Sep 28 '16 at 14:54
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    Related: Dirichlet's theorem on primes in arithmetic progression – Martin Sleziak Dec 26 '19 at 10:39

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