In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + ε)/ log 2) log log n and generalize it to the equation a 1!a 2! ... a k ! = n!. In a recent paper, F. Luca proved that n − b = 1 for large enough n provided that the ABC-hypothesis holds
The purpose of this note is to prove a counting Lemma from Peski [2015]. In the course of the proof,...
The left factorial of n is defined to be 0! + 1! + ··· + (n − 1)! and is denoted by !n. Kurepa conje...
AbstractLet p1,p2,… be the sequence of all primes in ascending order. The following result is proved...
In a paper published in 1993, Erdös proved that if n!=a! b!, where 1 < a≤b, then the difference b...
Let N0 denote the set of non-negative integers. In this paper we prove that lim sup t→∞ ∣∣{(n,m) ∈ ...
For a given positive integer n, we consider positive integers a(1), a(2)... at such that a(1) !a(2)!...
Let b ≥ 2 be an integer and denote by sb(m) the sum of the digits of the positive integer m when is ...
The proof of Theorem 1 uses lower bounds for linear forms in logarithms of algebraiC numbers (see [1...
summary:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
AbstractLet f(n) denote the number of factorizations of the natural number n into factors larger tha...
This paper discusses the difference between 0!=1 and 1!=1 for our understanding and also the usage o...
This document is a compilation of theorems on factorials, a brief proof of the relationship of facto...
Let n be a positive integer. We prove nn+1e−n 2π√ n − α ≤ n! < n n+1e−n2 2π√ n − β with the best ...
Under Baker's explicit abc conjecture, we completely solve a conjecture of Hickerson when a product ...
AbstractThe nth order difference [Δhn(x)m,g]x=a, where Δh is the difference operator with increment ...
The purpose of this note is to prove a counting Lemma from Peski [2015]. In the course of the proof,...
The left factorial of n is defined to be 0! + 1! + ··· + (n − 1)! and is denoted by !n. Kurepa conje...
AbstractLet p1,p2,… be the sequence of all primes in ascending order. The following result is proved...
In a paper published in 1993, Erdös proved that if n!=a! b!, where 1 < a≤b, then the difference b...
Let N0 denote the set of non-negative integers. In this paper we prove that lim sup t→∞ ∣∣{(n,m) ∈ ...
For a given positive integer n, we consider positive integers a(1), a(2)... at such that a(1) !a(2)!...
Let b ≥ 2 be an integer and denote by sb(m) the sum of the digits of the positive integer m when is ...
The proof of Theorem 1 uses lower bounds for linear forms in logarithms of algebraiC numbers (see [1...
summary:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
AbstractLet f(n) denote the number of factorizations of the natural number n into factors larger tha...
This paper discusses the difference between 0!=1 and 1!=1 for our understanding and also the usage o...
This document is a compilation of theorems on factorials, a brief proof of the relationship of facto...
Let n be a positive integer. We prove nn+1e−n 2π√ n − α ≤ n! < n n+1e−n2 2π√ n − β with the best ...
Under Baker's explicit abc conjecture, we completely solve a conjecture of Hickerson when a product ...
AbstractThe nth order difference [Δhn(x)m,g]x=a, where Δh is the difference operator with increment ...
The purpose of this note is to prove a counting Lemma from Peski [2015]. In the course of the proof,...
The left factorial of n is defined to be 0! + 1! + ··· + (n − 1)! and is denoted by !n. Kurepa conje...
AbstractLet p1,p2,… be the sequence of all primes in ascending order. The following result is proved...
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