Solving X^4+131=3y^4: No Integer Solutions

[dtl42]

This is a problem from Terence Tao's book, and I cannot locate the solution anywhere, so I thought I'd post it here.


1. Homework Statement

X^4+131=3y^4, Show that the equation has no solutions when x and y are integers.

Homework Equations

None? Not sure.

The Attempt at a Solution

I thought of using modular arithmetic, but I can't really work it correctly.

 

[Xevarion]

Okay, let's do some mods.

Mod (powers of) 2: 4th powers are 1 or 0 mod 16. 131 is 3 mod 16. So the LHS can be 3 or 4 mod 16 and the RHS can be 0 or 3 mod 16. Therefore we must have x even and y odd.

Mod 3: 4th powers are 1 or 0 mod 3. 131 is 2 mod 3. So the LHS can be 2 or 0 mod 3, and the RHS is 0 mod 3. So x cannot be divisible by 3.

Mod 5: 4th powers are 1 or 0 mod 5. 131 is 1 mod 5. So the LHS can be 1 or 2 mod 5, and the RHS can be 0 or 3 mod 5. Uh-oh! Looks like we're done!

Comments on strategy: When you're trying to solve a Diophantine equation, using mods is usually the place to start. If only one power appears, you might be able to guess what mod to use by Fermat's Little Theorem (in fact, in this case, this tells you to try 5, and that's the one that works). Powers of 2 and 3 are also good choices because there tend to be few possible residues mod those numbers. Using the given numbers can be useful too -- for example, it may also be possible to show that there are no solutions to this by looking mod 131. But I'm too lazy to compute all of the fourth powers mod 131...