September 27, 2025Open Access

The Diophantine equation x/y + y/x + z/w + w/z = n

Key Points

The diophantine equation has no solutions in positive integers under specific conditions, highlighting their mathematical significance.
Cases include conditions on n where prime factors are congruent to 3 modulo 4 or n is divisible by 8.
Elementary arguments were used for the prime factor condition, while brauer-manin obstruction applied for another case.
Understanding these conditions can advance research in number theory and the properties of diophantine equations.

Abstract

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.

The Diophantine equation x/y + y/x + z/w + w/z = n

Key Points

Abstract

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