Local obstructions and certified computations for perfect powers in power sums
Abstract
Let Sk(x)=1k+⋯+xk. We study perfect-power values of Sk(x) by combining local valuations with finite congruence certificates. For an irregular Bernoulli pair (p,r) with e=vp(Br)≥1, we prove a valuation formula uniform for k≡ r+1pmodpe(p−1), subject to an explicit condition on vp(x(x+1)). This gives explicit families of nonexistence results in which both x and k are unbounded. We also determine all solutions of Sk(x)=yn for 3≤ x≤106 with x≡0,3pmod4, leaving k and n≥2 unrestricted and including squares. This extends Hajdu's fixed-x classification in the same residue classes. A uniform restriction on the normalized odd part of Sk(x) and finite certificates exclude every k≥6; the remaining exponents are checked by exact integer arithmetic. The certificates admit independent verification. We further prove a limitation of congruence tests of bounded prime-power depth and solve the equation for every x≡28pmod5 110 664.
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BibTeX
@misc{pinter2026local,
title = {{Local obstructions and certified computations for perfect powers in power sums}},
author = {Ákos Pintér},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33567},
eprint = {2610.33567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33567},
}
RIS
TY - UNPB
TI - Local obstructions and certified computations for perfect powers in power sums
AU - Ákos Pintér
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33567
UR - https://arxiv.org/abs/2610.33567
AB - Let S_k(x)=1^k+⋯+x^k. We study perfect-power values of S_k(x) by combining local valuations with finite congruence certificates. For an irregular Bernoulli pair (p,r) with e=v_p(B_r)≥1, we prove a valuation formula uniform for k≡ r+1pmodp^e(p−1), subject to an explicit condition on v_p(x(x+1)). This gives explicit families of nonexistence results in which both x and k are unbounded. We also determine all solutions of S_k(x)=y^n for 3≤ x≤10^6 with x≡0,3pmod4, leaving k and n≥2 unrestricted and including squares. This extends Hajdu's fixed-x classification in the same residue classes. A uniform restriction on the normalized odd part of S_k(x) and finite certificates exclude every k≥6; the remaining exponents are checked by exact integer arithmetic. The certificates admit independent verification. We further prove a limitation of congruence tests of bounded prime-power depth and solve the equation for every x≡28pmod5 110 664.
ER -