Prime Numbers of the Form 24m+1 and Erdös-Straus Conjecture
Abstract
In 1948, Erdös and Straus formulated a conjecture on Egyptian fractions. One only needs to prove the conjecture for any prime number n such that n≡ 1 (mod 24). In this article, we give a necessary and sufficient polynomial condition for a number of the form 24m+1 to be a prime number. Then we present 56 solvability polynomials consisting of our earlier tame solutions, wild solutions and the polynomials for composite numbers of the form 24m+1. Numeric test shows that they imply the conjecture. Moreover, we give a list of 36 exponential-polynomial functions consisting of our earlier 28 tame solutions and the polynomials for composite numbers, whose numeric test shows that Howerton's conjecture on fourteen wild primes holds. Furthermore, we find a version of Goldbach conjecture modulo 24.
Related
- Erdős--Straus Conjecture: Modular Inversion in Mordell Classes
- A Ceiling Continued Fraction Approach to the Erdős-Straus Conjecture: Heuristic finiteness of counterexamples
- Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
- Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
- On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli
BibTeX
@misc{cao2026prime,
title = {{Prime Numbers of the Form $24m+1$ and Erdös-Straus Conjecture}},
author = {Yichuan Cao and Xiaoping Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32302},
eprint = {2610.32302},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32302},
}
RIS
TY - UNPB
TI - Prime Numbers of the Form 24m+1 and Erdös-Straus Conjecture
AU - Yichuan Cao
AU - Xiaoping Xu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32302
UR - https://arxiv.org/abs/2610.32302
AB - In 1948, Erdös and Straus formulated a conjecture on Egyptian fractions. One only needs to prove the conjecture for any prime number n such that n≡ 1 (mod 24). In this article, we give a necessary and sufficient polynomial condition for a number of the form 24m+1 to be a prime number. Then we present 56 solvability polynomials consisting of our earlier tame solutions, wild solutions and the polynomials for composite numbers of the form 24m+1. Numeric test shows that they imply the conjecture. Moreover, we give a list of 36 exponential-polynomial functions consisting of our earlier 28 tame solutions and the polynomials for composite numbers, whose numeric test shows that Howerton's conjecture on fourteen wild primes holds. Furthermore, we find a version of Goldbach conjecture modulo 24.
ER -