Solve a Diophantine equation with 2 variables and odd degree 5

Question

Solve in integers the equation: $a^{5} +1 = 2b^{5}$

Answer 1

There are no non-trivial solutions in positive integers to $a^p + 1 = 2b^p$ for any odd prime p.

This is a consequence of Denes Conjecture which states that if p is an odd prime and three natural non-zero integers $x^p, y^p$ and $z^p$ lie in an arithmetic progression, then x = y = z. Although referred to as a conjecture, this has been proved by Darmon & Merel. For further information and sources see section 9 and the references in this article by Ribet. Also useful is the English translation of item 13 in Ribet's references: Hellegouarch Y,, Invitation to the Mathematics of Fermat-Wiles, English Translation 2002, Academic Press, pp 342-3.

Clearly if $a^p + 1 = 2b^p$, then $1(= 1^p), b^p$ and $a^p$ are in arithmetic progression. Hence Denes Conjecture implies that $a = b = 1$, and so the only positive solution is $(a, b) = (1, 1)$.