Show that 3x2 7y2 1 has no integer solutionSolutionWork mo

Show that 3x² - 7y² = 1 has no integer solution.

Solution

Work modulo 7. The inverse of 3 is 5, so any solution x^2 is a solution of x^2 = 5(mod7)x^2. However, it is easy to check that 5 is not a quadratic residue of 7.

More simply, we can work modulo 3, since it is clear that the congruence (y^2) 1(mod3) does not have a solution.