31 problems

Erdős–Straus conjecture. For every integer $n\ge2$, there exist positive integers $x,y,z$ such that

Let $m\ge4$ be an integer, let $n$ be a positive integer, and call a rational number a sum of $m-1$ unit fractions if it can be written as … for positive integers…

Let $\frac{a}{b}$ be a positive rational number with squarefree denominator $b$. An Egyptian-fraction representation is a sum of distinct unit fractions whose denominators are prod…

Let $k$ be a positive integer, and consider the Egyptian-fraction equation … in positive integers $x,y,z$. Let $N_0$ denote the smallest integer from which the decomposition exists…

Let $m$ be fixed, and let a Type II solution refer to the representation type defined earlier in the paper for the equation governing representations of $m/p$ as a sum of three uni…

Let $m\ge20$ be an integer, and call a rational number a sum of three unit fractions if it can be written as $1/x+1/y+1/z$ with positive integers $x,y,z$. Prime-exception conjectur…

Let $m\ge4$ be an integer, and let a unit fraction mean a rational number with numerator $1$. Schinzel's conjecture. For every integer $m\ge4$, there exists a bound $n_m$ such that…

Breaking-point conjecture. The breaking point for the described denominator pattern at $F_3=257$ occurs somewhere around rank $R=186$. This is a computationally motivated predictio…

Mballa’s Conjecture. For every $n\geq 2$, there exist integers $x>\lfloor n/4\rfloor$ and $t\in\mathbb{N}^*$ such that

Mballa’s Conjecture. For every $n\in\mathbb{N}$ with $n\geq 2$, there exist $(x,t)\in(\mathbb{N}^*)^2$ and $q\in\mathbb{N}$ such that

Pseudo-greedy gap conjecture. If

Let $0<\theta\leq 1$ be rational, and let $R_n(\theta)$ denote the sum of the reciprocals in a best $n$-term Egyptian underapproximation of $\theta$. Eventually greedy underapproxi…

Let $m$ be a positive integer and let $n>1$. An Egyptian fraction representation is an expression of $\frac mn$ as a finite sum of unit fractions. Sierpiński's conjecture. For…

Let $m$ be a positive integer and let $n>1$. An Egyptian fraction representation is an expression of $\frac mn$ as a finite sum of unit fractions. Erdős's conjecture. For $m=4$, ev…

Let $u=\frac mn$ be a positive irreducible fraction. A faithful decomposition of $u$ is an expression … where the summands are positive rationals with distinct positive-integer den…

Let $x$ be a positive algebraic real number. Say that $x$ has the eventually greedy best Egyptian underapproximation property if, from some point onward, the greedy Egyptian-fracti…

Let $1\leq n\leq Q$ and let $k\geq 3$ be odd. A solution is a tuple $(a_1,q_1,\ldots,a_k,q_k)$ satisfying … with $\gcd(a_i,q_i)=1$, $|a_i|\leq n$, and $q_i\leq Q$ for every $i$. Ca…

Let $a\in\mathbb{N}\setminus\{1\}$, and let $b,c,d\in\mathbb{N}$ with $c\mid b+d$. The modulus is $4\operatorname{lcm}(b,d)$. Least-common-multiple formulation. For every…

Let $a$ be a natural number such that $w=4a+1$ is prime. The variables $r,v,s$ are natural numbers, and divisibility is in the integers. Prime $4a+1$ well-defined equation conjectu…

Let $k,n$ be positive integers such that $k/n\le 3$. The quantity $k/n$ is a positive rational number, and $\alpha,\beta,\gamma$ denote positive integers. Generalized three-unit-fr…

An Egyptian fraction representation of $1$ is a representation of the form … with positive integers as denominators. An integer $d$ is the second-largest denominator when it is the…

For fixed $k,m\in\mathbb{N}$, let $f_k(m,n)$ denote the number of representations … with $(a_1,\ldots,a_k)\in\mathbb{N}^k$ in increasing order. Subexponential representation conjec…

Burshtein's conjecture. There are other representations of $1$ of this form having $48$ terms, besides Allan Wm. Johnson's known example. The paper constructs new examples with…

Conjectured upper bound. The estimate

Let $n$ be a positive integer. A partition of $n$ into distinct squares is an expression … where $x_1,\ldots,x_k$ are distinct positive integers. Graham's conjecture. Every suffici…