I have a nice diophantine equation which I tried to solve since march but no solution. Tried modulo 11, tried to write it in some way to figure out a solution... I posted this a few months ago, but it was removed. The problem: Prove that the equation $x^2-y^{10}+z^5=6$ has no integer solutions( positive, negative). Maybe another idea if YOU have or a solution! I would appreciate it very much!
EDIT, Will Jagy: If $$ x \equiv 4,7; \; \; \; y \equiv 0; \; \; \; z \equiv 1,3,4,5,9 \pmod {11} $$ THEN $$x^2-y^{10}+z^5 \equiv 6 \pmod {11}$$ which you can just check. At the level of school mathematics, there are two tricks I know for this type of problem. Neither works, but other people have found dozens to hundreds of tricks that I don't know. I also know of published problems with no answer because the book screwed up. Apparently the answer will be published in late September.