On the prime power factorization of n

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. The Conjectures Bhargava factorials. For definitions and basic results dealing with Bhargava's factorial functions we refer to [3], [4], [5] and [8]. Briefly, let X ⊆ Z be a finite or infinite set of integers. Following [5], one can define the notion of a p-ordering on X and use it to define a set of generalized factorials of the set X inductively. By definition 0! X = 1. Whenever p a prime, we fix an element a 0 ∈ X and, for k ≥ 1, we select a k such that the highest power of p dividing k−1 i=0 (a k − a i) is minimized. The resulting sequence of a i is then called a p-ordering of X. As one can gather from the definition, p-orderings are not unique, as one can vary a 0. On the other hand, associated with such a p-ordering of X we define an associated p-sequence {ν k (X, p)} ∞ k=1 by ν k (X, p) = w p (k−1 i=0 (a k − a i)), where w p (a) is, by definition, the highest power of p dividing a (e.g., w 2 (80) = 16). One can show that although the p-ordering is not unique the associated p-sequence is independent of the p-ordering used. Since this quantity is an invariant it can be used to define generalized factorials of X by setting k! X = p ν k (X, p), (1) where the (necessarily finite) product extends over all primes p. Definition 1. [12]. An abstract (or generalized) factorial is a function ! a : N → Z + that satisfies the following conditions:

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. This is an unpublished updated version as of August 13, 2014.

Abstract. In this note, we make explicit approximation of the average of prime powers in the decomposition of n!. Then we find the order of geometric and harmonic means of such powers. ... 1.1. Approximate Formula for the Function Υ(n). First, we note that integrating by

This article presents a necessary and sufficient theorem for N numbers, coprime two by two, to be prime simultaneously

2020

Let P (n) be the largest prime factor of n. We give an alternative proof of the existence of infinitely many n such that P (n) > P (n + 1) > P (n + 2). Further, we prove that the set {P (n+1)/P (n)}n∈N has infinitely many limit points {0, xn, 1, yn}n∈N with 0 < xn < 1 < yn and limxn = lim yn = 1.

In this paper we collected problems, which was either proposed or follow directly from results in our papers.

Archiv der Mathematik, 2016

Let νp(n) be the exponent of p in the prime decomposition of n. We show that for different primes p, q satisfying some mild constraints the integers νp(n!) and νq(n!) cannot both be of a rather special form.

2007

We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.

Acta Arithmetica, 1995

Journal of Number Theory, 2008

Given integers k ≥ 2 and ℓ ≥ 3, let S * k,ℓ stand for the set of those positive integers n which can be written as n = p k 1 + p k 2 +. .. + p k ℓ , where p 1 , p 2 ,. .. , p ℓ are distinct prime factors of n. We study the properties of the sets S * k,ℓ and we show in particular that, given any odd ℓ ≥ 3, # ∞ k=2 S * k,ℓ = +∞.