Prove absolute convergence for higher exponents

Prove whether the remainder R(m,N) converges absolutely for every integer N\ge 4, where R(m,N) is the sum of 1/(x^N+y^N-z^N) over positive integers x,y,z with z\ne x and z\ne y.

Background

The analytical study indicates that the signed remainder R(m,N) converges for N\ge 4, and numerical evidence suggests that convergence becomes faster as N increases. However, convergence of the signed series does not imply absolute convergence, because positive and negative terms may cancel. The paper reports only preliminary numerical evidence for absolute convergence and leaves an analytical proof unresolved.

References

Similarly, we found that R(m,N) for N \ge 4 converges but we did not determine whether they converged absolutely.

— Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem  (2609.02112 - Edery, 2 Sep 2026) in Section 4, “Conclusion”