Sieve dimension and search depth for the Erdős-Straus conjecture, n ≡ 1 pmod24
Abstract
For primes n≡1pmod24, we study the depth of explicit, factorization-free searches for a decomposition of 4/n into three positive unit fractions. Let E2(N;J) be the set of such primes n≤ N for which no coprime pair 1≤ u,a≤ J admits a divisor s| an+u with s≡−1pmod4ua. For every fixed J, we prove |E2(N;J)|≪J N/(log N)1+A(J)/2, where A(J) counts the coprime pairs in [1,J]2. The proof uses a fixed-point-free involution and identifies the exact dimension of the covering. We also establish a dual Type I divisor criterion and an unconditional obstruction: at the fixed shift c=7, both branches of the cofactor criterion fail for asymp N/(log N)3/2 primes. At growing depth, an exact reciprocal-totient asymptotic with leading constant 9/(2π2) improves the explicit large-sieve bounds to O(Nexp(−0.0942(log N)2/3)) at cutoff p>J5, and to Oε(Nexp(−(0.1093−ε)(log N)2/3)) at cutoff p>J4+ε for fixed sufficiently small ε>0. Finally, combining a truncated Type I pass with a sparse residual search yields a complete, factorization-free batch decision procedure for all primes in [N,2N], using O(N2(log N)2exp(−c0(log N)2/3))=o(N2) arithmetic operations for some c0>0. The conjecture itself remains open.
Related
- Erdős--Straus Conjecture: Modular Inversion in Mordell Classes
- A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
- Prime Numbers of the Form 24m+1 and Erdös-Straus Conjecture
- Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
- Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
BibTeX
@misc{dahan2026sieve,
title = {{Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod24$}},
author = {Benjamin Dahan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.24035},
eprint = {2608.24035},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24035},
}
RIS
TY - UNPB
TI - Sieve dimension and search depth for the Erdős-Straus conjecture, n ≡ 1 pmod24
AU - Benjamin Dahan
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2608.24035
UR - https://arxiv.org/abs/2608.24035
AB - For primes n≡1pmod24, we study the depth of explicit, factorization-free searches for a decomposition of 4/n into three positive unit fractions. Let E_2(N;J) be the set of such primes n≤ N for which no coprime pair 1≤ u,a≤ J admits a divisor s| an+u with s≡−1pmod4ua. For every fixed J, we prove |E_2(N;J)|≪_J N/(log N)^1+A(J)/2, where A(J) counts the coprime pairs in [1,J]^2. The proof uses a fixed-point-free involution and identifies the exact dimension of the covering. We also establish a dual Type I divisor criterion and an unconditional obstruction: at the fixed shift c=7, both branches of the cofactor criterion fail for asymp N/(log N)^3/2 primes. At growing depth, an exact reciprocal-totient asymptotic with leading constant 9/(2π^2) improves the explicit large-sieve bounds to O(Nexp(−0.0942(log N)^2/3)) at cutoff p>J^5, and to O_ε(Nexp(−(0.1093−ε)(log N)^2/3)) at cutoff p>J^4+ε for fixed sufficiently small ε>0. Finally, combining a truncated Type I pass with a sparse residual search yields a complete, factorization-free batch decision procedure for all primes in [N,2N], using O(N^2(log N)^2exp(−c_0(log N)^2/3))=o(N^2) arithmetic operations for some c_0>0. The conjecture itself remains open.
ER -