Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T08:02:54Z http://mathoverflow.net/feeds/question/101258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101258/number-of-solutions-to-mx2ny2-equiv-k-pmodp Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$ Mike Decaro 2012-07-03T21:03:16Z 2012-07-03T23:55:02Z <p>I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in Gauss' Disquisitiones Arithmeticae, as it is proven from basic results in the theory of curves over finite fields. </p> <p>Is anyone aware of a specific reference where the number of solutions to the above congruence is discussed?</p> http://mathoverflow.net/questions/101258/number-of-solutions-to-mx2ny2-equiv-k-pmodp/101266#101266 Answer by GH for Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$ GH 2012-07-03T23:10:35Z 2012-07-03T23:10:35Z <p>A reference is my response <a href="http://mathoverflow.net/questions/65183/when-is-the-sum-of-two-quadratic-residues-modulo-a-prime-again-a-quadratic-residu/65309#65309" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/101258/number-of-solutions-to-mx2ny2-equiv-k-pmodp/101271#101271 Answer by Gerry Myerson for Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$ Gerry Myerson 2012-07-03T23:55:02Z 2012-07-03T23:55:02Z <p>Dickson's History, Volume II, page 286, says "G. Libri proved that there are $n\pm1$ sets of solutions $\lt n$ of $$x^2+ay^2+b\equiv0\pmod n$$ if $a,b$ are not divisible by the prime $n$." The reference is Jour. fur Math. 9 (1832) 182. </p> <p>Also, Dickson, page 296: "G. Frattini proved that the number of pairs of squares for which $x^2-Dy^2\equiv\lambda\pmod p$ is $(1/2)\{p-(D/p)\}$, where $(D/p)$ is the quadratic character of $D$ with respect to the prime $p$." Rendiconti Reale Accad. Lincei 4 (1885) 136-139. </p>