Thread: Another Diophantine Question

Hello community. Recently, I received a homework problem that has completely stumped me.

Show that the Diophantine equation: x^4 - 2y^4 = 1 has no nontrivial solutions. My hint is to consider congruences modulo 5.

Thank you for any assistance you may offer.

Originally Posted by qwath

Hello community. Recently, I received a homework problem that has completely stumped me.

Show that the Diophantine equation: x^4 - 2y^4 = 1 has no nontrivial solutions. My hint is to consider congruences modulo 5.

Thank you for any assistance you may offer.

To consider congruences modulo 5 is to consider all the remainders,





When we raise to the exponent of 4 the forms take respectively,




You might have realized that is Fermat's Little Theorem.

But anyways the two possible forms of (a fourth power) are: .

Thus, the possibilities are in the Diophantine Equation:
---> Impossible, RHS different form.
---> Impossible
---> Possible
---> Impossible.

Thus, what your homework was asking was not correct.
The second case, is still possible.