Egyptian Fractions with Restrictions

Abstract

Let $T_o(k)$ denote the number of solutions of ${\sum }_{i=1}^k\frac{1}{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1,p_2,...,p_t$, let $S(p_1, p_2,..., p_t)=\{p_1^{α_1}...p_t^{α_t}\mid α_i\in \mathbb{N}_0, i=1,2,..., t}$. Let $T_k(p_1,...,p_t)$ be the number of solutions ${\sum }_{i=1}^k\frac{1}{x_i}=1$ with $1<x_1<x_2<...<x_k$ and $x_i\in S(p_1,p_2,...,p_t)$. It is clear that if $T_k(p_1,...,p_t)\ne 0$ for some $k$, then the inverse sum of all elements $s_j>1$ in $S(p_1,p_2,...,p_t)$ is more than 1. In this paper we study $T_o(k)$ and $T_k(p_1,...,p_t)$. Three of our results are: 1) $T_o(2k+1)\ge (\sqrt{2}{)}^{(k+1)(k-4)}$ for all $k\ge 4$; 2) if the inverse sum of all elements $s_j>1$ in $S(p_1,p_2,...,p_t)$ is more than 1, then $T_k(p_1,...,p_t)\ne 0$ for infinitely many $k$ and the set of these $k$ is the union of finitely many arithmetic progressions; 3) there exists two constants $k_0=k_0(p_1,...,p_t)>1$ and $c=c(p_1,...,p_t)>1$ such that for any $k>k_0$ we have either $T_k(p_1,...,p_t)=0$ or $T_k(p_1,...,p_t)>c^k$.

18pages

Cited by in corpus (2)

• The number of Huffman codes, compact trees, and sums of unit fractions
• On solutions of ${\sum }_{i=1}^{n}1/x_i=1$ in integers of the form $2^a{k}^b$