Question

Prove that there are no solutions in integersxand yto the equation$x^2-5y^2=2$ .[Hint: Consider this equation modulo 5.]

Short answer

Original equation has no solution

Step-by-step solution

Step 1

$x^2-5y^2=2$.

Note that all squares end in or in decimal expansion.

$x^2-5y^2\equiv x^2\equiv 2mod5$.

If at all the original equation has solutions, then the equation above also has solutions.

But this implies that a square should be congruent to 2 modulo 5 which implies that the square should have the last digit in the decimal expansion as either 2 or 7 .

But such an integer does not exist.

Thus the original equation has no solution.