Related papers: Counting Egyptian fractions

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z| < 1\}$ normalized by $f (0) = 0$ and $f'(0) = 1.$ The logarithmic coefficients $\gamma_n$ of $f \in \mathcal{A}$ are…

In this paper, we show that there are only finitely many Narayana's numbers which can be written as product of three repdigits in base $g$ with $g \geq 2$. Moreover, for $2 \leq g \leq 10$, we determine all these numbers.

In this paper, we prove that for every integer $k \geq 1$, the $k$-abelian complexity function of the Cantor sequence $\mathbf{c} = 101000101\cdots$ is a $3$-regular sequence.

In this article, we count the quantity of minimal cyclic codes of length $n$ and dimension $k$ over a finite field $\mathbb F_q$, in the case when the prime factors of $n$ satisfy a special condition. This problem is equivalent to count the…

A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\{0,1\}^d$ whose vector sum cannot be generated as a positive integer linear combination of a proper subset of $S$. The largest possible…

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\| \ge 3\log_3 n$ for all $n$. Define the defect of…

Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak S}_n$. If the standard factorization of $M$ into primes is $M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}$, we define…

In this paper we examine the problem of computing majority function $\mathrm{MAJ}_n$ on $n$ bits by depth-two formula, where each gate is a majority function on at most $k$ inputs. We present such formula that gives the first nontrivial…

In number theory, the Erdos-Straus conjecture states that for all n >=2, the rational number 4/n can be expressed as the sum of three unit fractions. Paul Erdos and Ernst G. Straus formulated the conjecture in 1948. The restriction that the…

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three…

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg…

For each integer $t>0$ and each complex simple Lie algebra $\mathfrak{g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak{g}$ whose highest weight has height $t$. As a corollary, we classify…

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally…

Let $F$ be a field and let $E$ be an \'etale algebra over $F$, that is, a finite product of finite separable field extensions $E = F_1 \times \dots \times F_r$. The classical primitive element theorem asserts that if $r = 1$, then $E$ is…

We consider exchange of three intervals with permutation $(3,2,1)$. The aim of this paper is to count the cardinality of the set $3\iet(N)$ of all words of length $N$ which appear as factors in infinite words coding such transformations. We…

Given a finite group $G$ and positive integers $r$ and $s$, a problem of interest in algebra is determining the minimum cardinality of the product set $AB$, where $A$ and $B$ are subsets of $G$ such that $|A|=r$ and $|B|=s$. This problem…

This paper presents a means with time complexity of at worst O(n^3) to compute the discrete logarithm on cyclic finite groups of integers modulo p. The algorithm makes use of reduction of the problem to that of finding the concurrent zeros…

The notion of an Egyptian domain (where the analogue of Egyptian fractions works appropriately), first explored by Guerrieri-Loper-Oman, is extended to the more general notions of generically and locally Egyptian domains. Results from the…

We provide lower bounds on the number of subgroups of a group $G$ as a function of the primes and exponents appearing in the prime factorization of $|G|$. Using these bounds, we classify all abelian groups with 22 or fewer subgroups, and…