Related papers: A note on the Erd\H{o}-Straus Conjecture

This paper is a preliminary expository paper that outlines the relationship between solutions to the Erd\H{o}s-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses B\'{e}zout Coefficients…

In number theory, the Erdos-Straus conjecture states that for all n >=2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdos and Ernst G. Straus formulated the conjecture in 1948. The restriction that the…

In this article, we establish an additive decomposition of the discrete zeta function (for $s \in \mathbb{N}^*$, $s > 1$), more precisely of the function $4(\zeta(s)-1)$, as a series whose general term is of the form $1/x_n(s) + 1/y_n(s) +…

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\mathcal{O}_{\epsilon}(n^{3/5+\epsilon})$ solutions of…

It is an open question of Erd\H{o}s as to whether the alternating series $\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$ is (conditionally) convergent, where $p_n$ denotes the $n^{\mathrm{th}}$ prime. By using a random sifted model of the primes…

A strong version of Andrica's conjecture can be formulated as follows: Except for $p_n\in\{3,7,13,23,31,113\}$, that is $n\in\{2,4,6,9,11,30\}$, one has$\sqrt{p_{n+1}}-\sqrt{p_n} < \frac{1}{2}.$ While a proof is far out of reach I shall…

This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…

We develop a parametric approach to study the Diophantine equation $\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$, underlying the Erd\H{o}s--Straus ($k=4$), Sierpi\'nski ($k=5$), and related generalizations. We introduce and…

We find a polynomial in three variables whose values at nonnegative integers satisfy the Erd\H{o}s-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long…

Inspired by the proof of the Bertrand postulate given by P. Erd\H{o}S, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer…

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that…

A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum…

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding…

In 1948, Erd\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that…

In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$.…

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called…

A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erd\H{o}s conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the…

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\\i+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ where $B_n$ is the $n$-th…

The Erd\H{o}s primitive set conjecture states that the sum $f(A) = \sum_{a\in A}\frac{1}{a\log a}$, ranging over any primitive set $A$ of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved…

A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must…