Determine the distribution of cubic Fermat near misses

Determine whether the positive and negative cubic Fermat near misses satisfying x^3+y^3-z^3=+1 and x^3+y^3-z^3=-1, respectively, occur equally often among positive integers x, y, and z subject to zx and zy.

Background

The remainder R(m,3) contains terms associated with nontrivial solutions of x3+y3-z3=1. The paper argues that cancellation between positive and negative near misses is relevant to the expected logarithmic behavior of R(m,3), but notes that the required distributional result has not been established. Numerical data suggest that the difference between the numbers of +1 and −1 occurrences may remain small, yet this evidence does not resolve the question analytically.

References

There does not appear to be a theorem that has been proved that states that the +1 and -1 are equally distributed when x, y and z are positive integers with the constraint z\ne x and z\ne y.

— Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem  (2609.02112 - Edery, 2 Sep 2026) in Section 3.2, “Analytical study of convergence”