20 problems

Let $T_m=m(m+1)/2$ and let $E(8T_x+8T_y+T_z)$ be the set of positive integers not represented by $8T_x+8T_y+T_z$. Since $8T_x+1$ is an odd square, a sum of two odd squares and a tr…

Let $t_y=y(y+1)/2$ and $t_z=z(z+1)/2$ for $y,z\in\mathbb Z$. Sun's conjecture. Any positive integer $n$ is a sum of a square, an odd square and a triangular number; equivalently, e…

Conjecture. There exist rational numbers $a_{m,n}$ such that

Gurvich–Naumova conjecture. Every square pair $(k,j)$ is either one of the pairs $(k,j_i(k))$ for a positive integer $i$, or is a bi-Pythagorean pair of opposite parity.

Kane's conjecture. An arbitrary sum $\Delta_{\alpha_1,\dots,\alpha_k}$ represents all positive odd integers if and only if it represents each of

A triangular number is a number of the form $T_n=n(n+1)/2$ for a positive integer $n$, and a decimal representation consists of a single repeated digit when all its digits are equa…

Let $t_n=n(n+1)/2$ denote the $n$th triangular number, and let $g(a,b,c;p)$ denote the Frobenius number associated with the sequence of solutions parameterized by $p$. Conjecture.…

Sun Zhi-Wei's conjecture. For every integer $n>2$, neither $2^n+n$ nor $2^n-n$ is a triangular number.

For positive integers $a_1,\a_2,\a_3$, write $\Delta_{a_1,a_2,a_3}$ for the corresponding sum of triangular numbers; in particular, $\Delta_{1,4,5}$ is the ternary sum with coeffic…

For $n\in\mathbb{N}$ and positive integers $a,b,c$, let $N(a,b,c;m)$ be the number of integer solutions of $m=ax^2+by^2+cz^2$, and let $T(a,b,c;n)$ be the number of non-negative in…

Very triangular arithmetic-progression conjecture. For any positive integer $k$, there exists an arithmetic progression $AP(k)$ of length $k$ such that $t_j\in VT$ for all…

Twin very triangular number conjecture. There are infinitely many consecutive triangular numbers which are also very triangular.

Binary-weight conjecture. If $n\in\mathbb{N}$ is such that $|(n)_2^1|\geq 6$, then $|(t_n)_2^1|\geq 4$.

For positive integers $n$, let … where ${\mathrm F}(a_1,a_2,a_3)$ denotes the largest integer that is not a non-negative integer linear combination of the relatively prime positive…

Let $\mathbb{N}_0$ denote the set of nonnegative integers. Consider the two forms … and … where $a,b,c\in\mathbb{N}_0$. Three-form representation conjecture. Every natural number o…

Let $v_2(n)$ denote the exponent of $2$ in a positive integer $n$, and let $a,b,c$ be positive integers. Non-almost-universality conjecture. (i) If $v_2(b)\ge v_2(c)$,…

Let … where $T_z=z(z+1)/2$, and let $E(f)$ be the set of positive integers not represented by $f$. Sun's conjecture. The exceptional set is … The statement is presented as a comput…

For the four mixed forms … let $E(f)$ denote the set of positive integers not represented by $f$. Sun's conjecture. The exceptional sets are … … The source says that Kane confirmed…

For $k,l\in\mathbb{N}$, define the mixed quadratic-triangular form … where $T_z=z(z+1)/2$. Sun's conjecture. For every $k,l\in\mathbb{N}$, the form $2^kx^2+2^ly^2+T_z$ is almost un…

Let $T_x=x(x+1)/2$ for $x\in\mathbb{Z}$, let $a,b\in\mathbb{N}$, and let $r$ range over the positive odd integers. Sun's generalized conjecture. For any $a,b\in\mathbb{N}$ and odd…