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So I'm studying number theory, and my book said that
$$ x^2 + 5x + 7 \equiv 0 \pmod n $$ had at least four solutions for 1<n<20. I found four (n=3,7,13,19) but n=19 was tricky: I had to replace 7 with -50, which I assume was legit since $ 7 \equiv -50 \pmod {19} $ . Is this the only way to find the solution, by using trial and error to replace coefficients until you can complete the square?

Daniel
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  • See the linked dupes for how to solve modular quadratics. It's not clear what you mean by "at least four solutions". Please clarify. Better: state precisely what is claimed in your book instead of paraphrasing it. – Bill Dubuque Feb 15 '24 at 14:32
  • Are you familiar with quadratic reciprocity? If so you can use that to find the the prime moduli where the discriminant $-3,$ is a square, viz. $,p\equiv 1\pmod 3\ \ $ – Bill Dubuque Feb 15 '24 at 14:42
  • Noted that since $1\lt n\lt20$ we can solve easily this problem by brute force. – Piquito Feb 15 '24 at 15:29

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