Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem
Abstract
Consider the polynomial f=xN+yN−zN where x, y and z are positive integers and N ≥ 3 is an integer. By Fermat's Last Theorem, f is never zero so that its reciprocal, 1/f, has no singularities. We therefore study the finite sum of the reciprocal: S(m,N)=∑x=1m∑y=1m∑z=1m1/f. The terms 1/f can be positive, negative and their magnitude is less than unity. A key observation is that S(m,N) can be split into two convenient parts: a dominant contribution D(m,N) that has a simple analytical expression and a remainder R(m,N) which is more complicated but negligible compared to D(m,N). Therefore, S(m,N) is almost identical to D(m,N). The analytical expression for D(m,N) is (2 m−1) Hm(N) where Hm(N)=∑x=1m1/xN approaches quickly the Riemann zeta function ζ(N) as m increases. Therefore, the original sum S(m,N) has a simple expression: it is basically linear in m with slope equal to 2 ζ(N). Its linear behavior is not an asymptotic result; plots of S(m,N) vs. m for different N show a straight line starting at m=1. S(m,N) deviates slightly from a straight line over a small interval 8≤ m≤ 12 for the case N=3. This slight deviation is due to Fermat near misses where x3+y3−z3=± 1 (for z≠ x and z≠ y); these create a jump in the remainder R(m,3) at m=9. We make a numerical and analytical study of the remainder R(m,N). From the numerical analysis, R(m,N) converges for N≥ 4 but it was harder to tell whether N=3 converged. An analytical study based on a comparison of R(m,N) to its Cauchy principal value integral, shows that R(m,3) likely diverges logarithmically. It also shows that R(m,N) converges for N≥ 4 in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.
Related
- Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
- Jacob's ladders and wander from Hardy-Littlewood integral (1918) to the ζ-Platonic cube form of equivalent of the Fermat-Wiles theorem
- Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
- Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones
- Further bounds in the polynomial Szemerédi theorem
BibTeX
@misc{edery2026summing,
title = {{Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem}},
author = {Ariel Edery},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02112},
eprint = {2609.02112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02112},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem
AU - Ariel Edery
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.02112
UR - https://arxiv.org/abs/2609.02112
AB - Consider the polynomial f=x^N+y^N−z^N where x, y and z are positive integers and N ≥ 3 is an integer. By Fermat's Last Theorem, f is never zero so that its reciprocal, 1/f, has no singularities. We therefore study the finite sum of the reciprocal: S(m,N)=∑_x=1^m∑_y=1^m∑_z=1^m1/f. The terms 1/f can be positive, negative and their magnitude is less than unity. A key observation is that S(m,N) can be split into two convenient parts: a dominant contribution D(m,N) that has a simple analytical expression and a remainder R(m,N) which is more complicated but negligible compared to D(m,N). Therefore, S(m,N) is almost identical to D(m,N). The analytical expression for D(m,N) is (2 m−1) H_m^(N) where H_m^(N)=∑_x=1^m1/x^N approaches quickly the Riemann zeta function ζ(N) as m increases. Therefore, the original sum S(m,N) has a simple expression: it is basically linear in m with slope equal to 2 ζ(N). Its linear behavior is not an asymptotic result; plots of S(m,N) vs. m for different N show a straight line starting at m=1. S(m,N) deviates slightly from a straight line over a small interval 8≤ m≤ 12 for the case N=3. This slight deviation is due to Fermat near misses where x^3+y^3−z^3=± 1 (for z≠ x and z≠ y); these create a jump in the remainder R(m,3) at m=9. We make a numerical and analytical study of the remainder R(m,N). From the numerical analysis, R(m,N) converges for N≥ 4 but it was harder to tell whether N=3 converged. An analytical study based on a comparison of R(m,N) to its Cauchy principal value integral, shows that R(m,3) likely diverges logarithmically. It also shows that R(m,N) converges for N≥ 4 in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.
ER -