Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.11672

The Erd\"os-Straus Conjecture and Pythagorean Primes

Bernd R. Schuh
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on allowed Pythagorean primes. Empirical evidence suggests that these restrictions hold for all Pythagorean primes, which I state as two independent conjectures. One can be formulated as follows: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4abb/d, where a, b, c are natural numbers and d is a divisor of ab. The second conjecture reads: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4ac, where a, b, c are natural numbers. I give a new straightforward plausibility for the latter conjecture (which has been formulated independently by other authors) and I outline a practicle and effective algorithm to determine a,b,c for a given p.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Bernd R. Schuh. 2025-02-26. The Erd\"os-Straus Conjecture and Pythagorean Primes. https://arxiv.org/abs/2503.11672

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗