# Solve another Diophantine equation with 2 variables and odd degree 5

Solve a Diophantine equation with 2 variables and odd degree 5

Prove that there are no non trivial integer solutions to the equation $a^{5} -1 = 2b^{5}$

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why do you care?? – jspecter Jan 17 '13 at 20:18
jspecter said : "why do you care??" I care because i want to understand. If nobody cared, what would happen ? we have to care for better. – user55514 Jan 23 '13 at 7:45

Let $a$ and $b$ satisfy $a^5 - 1 = 2b^5.$ Then $A = -a$ and $B = -b$ satisfies $A^5 + 1 = 2B^5.$ So by your previous question...
I have edited my answer to the original question as I had overlooked that Denes' Conjecture relates only to positive integers. It seems therefore to remain possible that there are non-trivial solutions to $a^5 + 1 = 2b^5$ with a, b negative. I am not sure therefore that the above answer resolves $a^5 - 1 = 2b^5$, although it is a correct inference if it is assumed that $a^5 + 1 = 2b^5$ has no non-trivial solutions in positive or negative integers. – Adam Bailey Jan 18 '13 at 9:41