Let $$S=\left\langle s_1,\ldots ,s_n\right\rangle $$ be a numerical semigroup generated by the relatively prime positive integers $$s_1,\ldots ,s_n$$ . $$k\geqslant 2$$ an integer. In this paper, we consider following k-power variant of Frobenius number S defined as $$\begin{aligned} {}^{k\!}r\!\left( S\right) := \text { largest } k {-power integer not belonging to S. \end{aligned}$$ investigat...

In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal t...

In this paper, we find a formula for the largest integer (p-Frobenius number) such that linear equation of non-negative coefficients composed Jacobsthal triplet has at most p representations. For p=0, problem is reduced to famous Diophantine Frobenius, which called Frobenius number. We also give closed number integers (p-genus), equations have Extensions polynomial and Jacobsthal–Lucas more gen...

Let $0<\lambda\leq1$, $\lambda\notin\left\{\frac24, \frac27, \frac2{10}, \frac2{13}, \ldots\right\}$, be a real and $p$ prime number, with $[p,p+\lambda p]$ containing at least two primes. Denote by $f_\lambda(p)$ the largest integer which cannot written as sum of primes from p]$. Then \[f_\lambda(p)\sim\left\lfloor2+\frac2\lambda\right\rfloor\cdot p\text{, }p\text{ goes to infinity.}\] Further...

let $fneq1,3$ be a positive integer‎. ‎we prove that there exists a numerical semigroup $s$ with embedding dimension three such that $f$ is the frobenius number of $s$‎. ‎we also show that‎ ‎the same fact holds for affine semigroups in higher dimensional monoids‎.

Given a primitive integer vector a ∈ Z>0, the largest integer b such that the knapsack polytope P = {x ∈ R≥0 : 〈a,x〉 = b} contains no integer point is called the Frobenius number of a. We show that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number. More precisely, we prove that it does not essentially exceed ||a|| 1+...

Iskander Aliev A sharp lower bound for the Frobenius number Ferdinand Georg Frobenius (1849–1917) raised the following problem: given N positive integers a1, . . . , aN with gcd(a1, . . . , aN ) = 1, find the largest natural number gN = gN (a1, . . . , aN) (called the Frobenius