Related papers: Short Proof: Exact Solution to the Finite Frobeniu…
We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact…
A partition into distinct parts is refinable if one of its parts $a$ can be replaced by two different integers which do not belong to the partition and whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of being…
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound $s$ on the number of distinct primes dividing the…
In this paper, we will give an improvement on the lower bound for the counterfeit coins problem in the case that the number of false coins is unknown in advance
In a base phi representation a natural number is written as a sum of powers of the golden mean $\varphi$. There are many ways to do this. How many? Even if the number of powers of $\varphi$ is finite, then any number has infinitely many…
Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer not belonging to the set $\big\{ \sum_{i=1}^na_ix_i\ |x_i\in \mathbb{N}\big\}$. The general Frobenius…
A number $\alpha$ has a representation with respect to the numbers $\alpha_1,...,\alpha_n$, if there exist the non-negative integers $\lambda_1,... ,\lambda_n$ such that $\alpha=\lambda_1\alpha_1+...+\lambda_n \alpha_n$. The largest natural…
The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity…
Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive…
In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…
We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…
How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We…
We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of…
The additive square problem is a relatively famous open problem in the area of combinatorics on words: Does there exist an infinite word over a finite alphabet, such that no two consecutive blocks of the same length have the same sum? In…
This paper considers the arbitrary-proportional finite-set-partitioning problem which involves partitioning a finite set into multiple subsets with respect to arbitrary nonnegative proportions. This is the core art of many fundamental…
The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of 5 cubes of natural numbers. We also conjecture that any even perfect number, other than $6$, can be written as the sum of only 3…
Let $n$ be a positive integer greater than $2$. We define \textit{the Proth numerical semigroup}, $P_{k}(n)$, generated by $\{k 2^{n+i}+1 \,\mid\, i \in \mathbb{N}\}$, where $k$ is an odd positive number and $k < 2^{n}$. In this paper, we…
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…
Given a set of three positive integers {a1, a2, a3}, denoted A, the Frobenius problem in three variables is to find the greatest integer which cannot be expressed in the following form, where x1, x2 and x3 are non-negative integers: x1*a1 +…