Add to ORKGLet N0 denote the set of non-negative integers. In this paper we prove that lim sup t→∞ ∣∣{(n,m) ∈ N20: n!m! = t}∣ ∣ = 6. 1
AbstractLet p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the n...
This document is a compilation of theorems on factorials, a brief proof of the relationship of facto...
This paper presents an effective method for calculating !n, and implements the Smarandache-Kurepa fu...
In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference ...
summary:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
AbstractIt has long been observed that certain factorization algorithms provide a way to write the p...
AbstractIf n is a positive integer, we write n! as a product of n prime powers, each at least as lar...
This paper presents a binomial theorem and proof based on Annamalai’s binomial identity. The factori...
For any positive integer k let φ(k), σ(k), and τ(k) be the Euler function of k, the divisor sum func...
Our goal is to prove the following asymptotic estimate for n!, called Stirling’s formula. Theorem 1....
AbstractFor a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For t...
This paper presents the innovative binomial theorems and proofs based on Annamalai’s binomial identi...
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 ...
For a given positive integer n, we consider positive integers a(1), a(2)... at such that a(1) !a(2)!...
Abstract. The summation formula n−1∑ i=0 εii!(ik + uk) = vk + ε n−1n!Ak−1(n) (ε = ±1; k = 1, 2,...;...
AbstractLet p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the n...
This document is a compilation of theorems on factorials, a brief proof of the relationship of facto...
This paper presents an effective method for calculating !n, and implements the Smarandache-Kurepa fu...
In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference ...
summary:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
AbstractIt has long been observed that certain factorization algorithms provide a way to write the p...
AbstractIf n is a positive integer, we write n! as a product of n prime powers, each at least as lar...
This paper presents a binomial theorem and proof based on Annamalai’s binomial identity. The factori...
For any positive integer k let φ(k), σ(k), and τ(k) be the Euler function of k, the divisor sum func...
Our goal is to prove the following asymptotic estimate for n!, called Stirling’s formula. Theorem 1....
AbstractFor a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For t...
This paper presents the innovative binomial theorems and proofs based on Annamalai’s binomial identi...
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 ...
For a given positive integer n, we consider positive integers a(1), a(2)... at such that a(1) !a(2)!...
Abstract. The summation formula n−1∑ i=0 εii!(ik + uk) = vk + ε n−1n!Ak−1(n) (ε = ±1; k = 1, 2,...;...
AbstractLet p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the n...
This document is a compilation of theorems on factorials, a brief proof of the relationship of facto...
This paper presents an effective method for calculating !n, and implements the Smarandache-Kurepa fu...