Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
Abstract
In 1948, Erdös and Straus formulated a conjecture : for any positive integer n>2, there exist positive integers n1,n2 and n3 such that 4/n=1/n1+1/n2+1/n3,nonumber which is still open. It is known that the conjecture holds if one can prove it for any prime n≡ 1 (mod 24). If n=24m+1 and n1≤ n2,n3, then n1=6m+k with 1≤ k≤ 12m. A solution (n1,n2,n3) of the above equation is called a {\it tame solution} if n2 and n3 are factors of (6m+k)(24m+1). We call n=24m+1 {\it wild} if it does not have any tame solution. Based on the information in our earlier work on tame solutions posed in arXiv, Howerton found that there are only fourteen wild primes of the form n=24m+1≤ 2.4× 1011. In this paper, we derive thirty-four families of wild solutions of the above equation, which contain the solvability of the fourteen wild primes. Together with our earlier tame polynomial solutions, numeric test shows that they cover all the primes of the form 24m+1.
Related
- Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
- Erdős--Straus Conjecture: Modular Inversion in Mordell Classes
- ON THE EXISTENCE OF SOLUTIONS TO ERDŐS-STRAUS TYPE EQUATIONS
- 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
BibTeX
@misc{xu2026congruence,
title = {{Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions}},
author = {Xiaoping Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29250},
eprint = {2609.29250},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29250},
}
RIS
TY - UNPB
TI - Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild Solutions
AU - Xiaoping Xu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29250
UR - https://arxiv.org/abs/2609.29250
AB - In 1948, Erdös and Straus formulated a conjecture : for any positive integer n>2, there exist positive integers n_1,n_2 and n_3 such that 4/n=1/n_1+1/n_2+1/n_3,nonumber which is still open. It is known that the conjecture holds if one can prove it for any prime n≡ 1 (mod 24). If n=24m+1 and n_1≤ n_2,n_3, then n_1=6m+k with 1≤ k≤ 12m. A solution (n_1,n_2,n_3) of the above equation is called a {\it tame solution} if n_2 and n_3 are factors of (6m+k)(24m+1). We call n=24m+1 {\it wild} if it does not have any tame solution. Based on the information in our earlier work on tame solutions posed in arXiv, Howerton found that there are only fourteen wild primes of the form n=24m+1≤ 2.4× 10^11. In this paper, we derive thirty-four families of wild solutions of the above equation, which contain the solvability of the fourteen wild primes. Together with our earlier tame polynomial solutions, numeric test shows that they cover all the primes of the form 24m+1.
ER -