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Is there an elementary way of showing the infinitude of primes of the form $5n+4$ without using quadratic reciprocity?

Alonso Babuhicrik
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  • Without using quadratic reciprocity see here. Actually, quadratic reciprocity is more elementary, right? – Dietrich Burde Feb 25 '19 at 17:51
  • I also said elementary:) – Alonso Babuhicrik Feb 25 '19 at 17:52
  • I was thinking like this... For any $N=5(n!)^2-1$ if I could show that there is a prime $p$ of the form greater than $n$ then I'm done. Its easy to eliminate the the case that $p$ isn't less than or equal to $n$ and also not of the form $5k+1$. But then I got stuck... – Alonso Babuhicrik Feb 25 '19 at 17:56
  • Do you want a proof like this? https://math.stackexchange.com/questions/244915/infinitely-number-of-primes-in-the-form-4n1-proof – Seewoo Lee Feb 25 '19 at 17:57
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    In Nagell's book on number theory, there's an elementary proof of the infinitude of primes $\equiv-1\pmod N$ for each $N$. I don't think it uses quadratic reciprocity. – Angina Seng Feb 25 '19 at 18:01
  • I think a proof by quadratic reciprocity is really the best, and moreover completely elementary. – Dietrich Burde Feb 25 '19 at 19:09
  • @ArpanDas, this question is related to famous Dirichlet theorem about primes that says: Binomial $ax+b$ where (a, b)=1, gives infinitely many primes. – sirous Mar 01 '19 at 16:30

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I wanted to give this as a comment but it was too long. For a proof one must consider following points:

*n must be odd and $5n+4$ results a set of infinite numbers of form $...9=10 x +9$:

$9, 19, 29, 39, 49, 59, 69, 79, . . .$

  • This set has common members with the set of primes resulted by known generating functions,in fact if n is large enough $5n+4$ can always be transformed to a polynomial identical to one form of generating function, here is some example:

$79=2\times 3\times 13 +1$ (Euclidean theorem for primes)

$59=2\times 29+1$, $ 19=4\times 5 -1 $, $149=6\times 25 -1$, represent samples of Dirichlet primes resulted from $2x+1$, $4x-1$ and $6x-1$ respectively.

$229=2\times 10^2 +29$ represent a sample of $2x^2+29$ which is Euler's function.

$1129=30\times 37 + 19$ which represent a theorem that says all primes are of form $30 x+r$ where $r=1, 7, 11, 13, 19,23, 29 $ particularly when $r=19$ and $r=29$; in other words primes resulted from $5n+4$ are members of a subset of primes of form $30 x +19$ or $30 x+29$.

sirous
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