The equation in question is x*x≡1097 (mod 65539).
The online calculators and sagemath have said there are no solutions but I can work it out on paper using the Legendre properties.
How come sagemath doesn't think there is a solution?
2026-04-05 23:19:52.1775431192
I am trying to understand whether this equation has a solution using either Legendre's or Jacobi's symbols
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You can show that it has no solutions using the Legendre symbol:
$\begin{eqnarray}\left( \frac{1097}{65539} \right ) & = & \left( \frac{65539}{1097} \right) \times (-1)^{\frac{1097-1}{2}\frac{65539-1}{2}} \\ & = & \left( \frac{816}{1097} \right) \\ & = & \left( \frac{2^4}{1097} \right) \left( \frac{3}{1097} \right) \left( \frac{17}{1097} \right) \\ & = & 1 \times \left( \frac{1097}{3} \right) \left( \frac{1097}{17} \right) \\ & = & \left( \frac{2}{3} \right) \left( \frac{9}{17} \right) \\ & = & -1 \times 1 \\ & = & -1 \end{eqnarray}$
Which shows that 1097 is not a quadratic residue modulo 65539.