## Conditions for Quadratic Diophantine Equation to Have Solutions

\[x^2-ny^2= \pm 1\]

does not always have solutions. If \[n \equiv 3 \; (mod \; 4)\]

then working modulus 4 and using the fact that the only quadratic residue modulus 4 is 1 - so that \[x^1 \equiv 1 \; (mod \; 4)\]

for every integer \[x\]

we have \[x^2 -ny^2 \equiv 1-3 \times 1 \; (mod \; 4) =-2 \; (mod \; 4)\]

so no solutions exist. If solutions exist for the equation

\[x^2-ny^2=m\]

then working modulus \[d\]

where \[d\]

is any divisor of \[n\]

we have \[x^2 \equiv m \; (mod \; d)\]

so \[m\]

must be a quadratic residue of \[d\]

.The converse is not true - that is if

\[m\]

is a quadratic residue of \[d\]

that solutions exist.Consider the equation

\[x^2-34y^2=-1\]

. Working modulus 17 \[x^2 \equiv -1 \; (mod \; 17) \equiv 16 \; (mod \; 17)\]

and 16 is a quadratic residue of 17 but the equation has no solutions because the continued fraction for \[\sqrt{34}= [ 5 \lt 1,4,1, 10 gt ]\]

contains a cycle of even length.