14 problems

Let $f(a,b,c)=g(a,b,c)+a+b+c$ be the shifted three-variable Frobenius number for relatively prime positive integers $a,b,c$. Empirical upper-bound conjecture. … This conjecture is…

Let $(a,b,c)$ be an admissible triplet, meaning a triplet of positive integers satisfying the admissibility conditions for the three-variable Frobenius problem, and let $f(a,b,c)$…

Let $a_1,a_2,a_3$ be relatively prime positive integers, and let the Frobenius number be the largest integer $t$ such that … has no solution in nonnegative integers $m_1,m_2,m_3$.…

Let $a$ be an integer with $a>30$, and let … Let $S(A(a))$ be the numerical semigroup generated by $A(a)$. For $n\in\mathbb{Z}^{+}$, let $\iota(n)$ be the least number of positive…

Let $g(k)$ be the least positive integer such that for every pair of integers satisfying $g(k)<a_1<a_2$, there is a prime power $p^k\le g_{a_1,a_2}$ of the form $p^k=a_1x+a_2y$ wit…

Let $a_1$ and $a_2$ be integers with $a_2>a_1>40$ and $\gcd(a_1,a_2)=1$. Let $\pi_{2,a_1,a_2}$ denote the number of prime squares $p^2\le g_{a_1,a_2}$ representable as…

Unboundedness conjecture. For every positive integer $m\geqslant 1$, there exists some $\alpha\in\mathfrak{O}_K^+$ such that

Let $s\ge 3$. Let $a_k$, where $k\in [s]$, be positive integers such that $(a_1,a_2,\ldots,a_s)=1$. Then $S_{s,s-2}^m(a_1,a_2,\ldots,a_s)$ denotes the set of positive integers sati…

Conjecture. For any solution $(\hat x,\hat y,\hat z)$ of $px+qy+lz=n$ with $\gcd(p,q,l)=1$, there exist solutions $s_i$, $i=1,2,3$, of either $qy+lz=n$, $px+lz=n$, or $px+qy=n$ suc…

Let $\\{S_t:t\in\\N\\}$ be a parametric Presburger family, and suppose $S_t$ is finite. Write $\max(S_t)$ for its maximum. Woods's parametric Frobenius conjecture. The function…

Parametric relation-set conjecture. There exists a PILP whose lattice point set equals $G(S(t))$ for $t\gg0$. The same is true for

Roune–Woods conjecture. The function $D$ is an eventual quasi-polynomial (EQP).

Let $a_i:bbZ_+crightarrowbbZ_+$, $1cle icle n$, be eventually positive functions in $cmathrm{EQP}$, where $F(a_1,cdots,a_n)$ denotes the largest element of the group…

Let $a$, $b$, and $c$ be integers with $(a,b,c)=1$, and let $f(a,b,c)$ denote the greatest integer that cannot be represented as a positive linear combination of $a$, $b$, and $c$.…