20 problems

Let $a_1,a_2,a_3$ be the components of an input vector for a three-variable integer knapsack, and let $\mathrm{F}(\boldsymbol a)$ denote its Frobenius number. Exclude the special f…

Arnold's conjecture. For a typical vector $a$ with $\vert a\vert_1=T$, the Frobenius number grows like

Let $a,b,c,\ell,r,q,u$ and $\mu'$ be as defined above, and let $\mathcal{X}_{dist}$ be the set of distances between consecutive elements of the ordered set $\mathcal{X}_{ord}$. Sup…

Large-shift infinite square-sequence conjecture. If $a>30$, then

Einstein et al.'s conjecture. The Frobenius number of this sequence is given by

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.…

Let $r_2(m,n)$ denote the largest perfect square that cannot be expressed as $mx+ny$ for nonnegative integers $x$ and $y$, and let $(u_n)$ be the sequence defined by $u_1=1$,…

Let $p$ be a prime, let $\lambda>0$, let $I_\lambda(p)=[p,p+\lambda p]$, and let $S_\lambda(p)$ be the numerical semigroup generated by the primes in $I_\lambda(p)$, with Frobenius…

Let $p$ be a prime, let $(K,\mathcal{O},k)$ be a $p$-modular system, and let $B$ be a block of $\mathcal{O}G$ for a finite group $G$. The strong $\mathcal{O}$-Frobenius number…

Telescopicity reversal conjecture. The first sequence is telescopic if and only if the second sequence is not telescopic. Consequently, the Frobenius number of the first sequence s…

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 $R_n^k$, with $n=T_km+j$, be a numerical semigroup given by its minimal relations on the residue class of $n$ modulo $T_k$: … If $k=2q$, the polynomials $E_{ij}^{(k)}(m)$ have…

Let $g(a_1,\ldots,a_n)$ denote the Frobenius number of the tuple of positive integers $(a_1,\ldots,a_n)$. For integers $k\geqslant 5$, consider the tuple…

Let $a_1<\cdots<a_d$ be coprime positive integers, and let $F$ be the largest integer that is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote t…

Let $a_1(t),\ldots,a_d(t)$ be polynomial functions of $t$ that are eventually positive, and let the Frobenius number be the largest integer not belonging to the semigroup generated…

Let $(a,b,c)$ be a vector of three positive integers with $\gcd(a,b,c)=1$, and let $F(a,b,c)$ denote its Frobenius number, the smallest integer $f_0$ such that every integer…

Frobenius discrepancy conjecture for three generators.

Let $(p,q)$ be a pair of primes, and let $g$ denote the relevant Frobenius number and $G_2$ the established upper bound for the associated representation problem. The bound…

Let $n$ be a square-free odd integer with $s\geq 1$ distinct prime factors. Let $\nu_n$ denote the largest non-genus of a ${\mathbb Z}_n$ action, and let $g_n$ be the Frobenius num…

Let ${\boldsymbol a}=(a_1,\ldots,a_N)\in\mathbb Z_{>0}^N$ satisfy $\gcd({\boldsymbol a})=1$, and define … Let $g_N({\boldsymbol a})$ be the Frobenius number, put…