On the Diophantine equation $x^4+y^4=c$
Volume 204 / 2022
Acta Arithmetica 204 (2022), 141-150
MSC: Primary 11D25; Secondary 11R27, 14G05.
DOI: 10.4064/aa210718-5-4
Published online: 6 June 2022
Abstract
We study the equation \begin{equation} x^4+y^4=c, \tag{$*$} \end{equation} where $c \leq 10^4$ is a positive integer. First, we show that when $c=7537$ or $c=8882$ then $(*)$ has no rational solutions, hence completing Henri Cohen’s table on the solvability of $(*)$ in the rational numbers. Second, for all $c\leq 10^4$, where $(*)$ is everywhere locally solvable but globally unsolvable, we show that for any positive integer $d\geq 2$, equation $(*)$ has solutions in some number field of degree $d$.
