Is there integer $x$ such that $79|7x^2+4x-23$

Question

Is there integer $x$ such that $79|7x^2+4x-23$ ?

I keep getting that there is $x$ that satisfies this condition, but online calculator keeps saying that there is not. I worked it out using Legendre's symbol:

If $y=7x+2$, then starting equation is equivalent to $y^2 \equiv 7$ mod$79$, and because $\genfrac{(}{)}{}{}{7}{79} = \genfrac{(}{)}{}{}{79}{7} = 1$, equation has a solution ?

Possible duplicate of Quadratic reciprocity: Tell if $c$ got quadratic square root mod $p$ — Bill Dubuque, Jan 25 '19 at 15:44

studiosus · Answer 1 · 2019-01-25T15:11:27.430

1

Note that by quadratic reciprocity $$ \left(\frac{7}{79}\right)=(-1)^{3\cdot39}\left(\frac{79}{7}\right)=-\left(\frac{2}{7}\right)=-1. $$

edited Jan 25 '19 at 15:11

answered Jan 25 '19 at 15:09

studiosus

4,147

Oh my god ..... – Jan 25 '19 at 15:10

Is there integer $x$ such that $79|7x^2+4x-23$

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