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Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame 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 one only needs to prove the conjecture for any prime number n such that 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. In this paper, we derive twenty-eight families of tame solutions of the above equation. Numeric test shows that they cover all the tame primes.
Related
- Congruence Classes of Supporting the Erdös-Straus Conjecture II: Wild 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 I: Tame Solutions}},
author = {Xiaoping Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.23601},
eprint = {2605.23601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.23601},
}
RIS
TY - UNPB
TI - Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions
AU - Xiaoping Xu
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2605.23601
UR - https://arxiv.org/abs/2605.23601
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 one only needs to prove the conjecture for any prime number n such that 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. In this paper, we derive twenty-eight families of tame solutions of the above equation. Numeric test shows that they cover all the tame primes.
ER -