Solving Modulo a Prime: x^3 + 2y^3 =5

[ksk104]

Homework Statement

Show the equation x^3 + 2y^3 =5 has no solution for x,y in Z, by considering it modulo a prime

Homework Equations

The Attempt at a Solution

I need help starting this problem, I've been stuck on it for a while and don't even have a clue of how to start it.
Thanks

 

[HallsofIvy]

When it says "modulo a prime" it does not mean you must prove it has no solutions modulo any prime. Just try a convenient prime. In particular, consider this modulo 5: it reduces to x³+ 2y[/sup]3[/sup]= 0. You could just do the calculations for all 25 pairs from (0,0) to (4,4) but I think you can just note that x³ and 2y³ are additive inverses. The additive inverse of 1 is 4 and the additive inverse of 2 is 3. Can one be x³ and the other 2y³? Of course, x= y= 0 does solve this equation (modulo 5) which tells you that any solution to the original equation must involve only multiples of 5.