This paper demonstrates the absence of solutions for certain n conditions in positive integers, indicating important number theory implications.
This paper shows that the equation in the title has no solutions in the positive integers if (i) n has prime factor congruent to $3$ (modulo $4$); or (ii) $8|n$; or $8|n-1$. Case (i) follows from an elementary argument. Case (ii) follows from the previous works of E. Dofs and the second author. Case (iii) follows from a standard Brauer-Manin obstruction argument.
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Tân et al. (2025) studied this question.
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