𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On the Parity of Exponents in the Factorization ofn!

✍ Scribed by Daniel Berend

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
264 KB
Volume
64
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

It is shown that, for any k, there exist infinitely many positive integers n such that in the prime power factorization of n!, all first k primes appear to even exponents. This answers a question of Erdo s and Graham (``Old and New Problems and Results in Combinatorial Number Theory,'' L'Enseignement Mathe matique, Imprimerie Kundia, Geneva, 1980). A few generalizations are provided as well.

1997 Academic Press

Question. Does there exist, for every fixed k, some n>1 with all the exponents _ 1 (n), _ 2 (n), ..., _ k (n) even?

Our first result answers this question in the affirmative.

πŸ“œ SIMILAR VOLUMES

On the Parity of Exponents in the Prime
On the Parity of Exponents in the Prime Factorization of Factorials
✍ J.W. Sander πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 126 KB

In 1997 Berend proved a conjecture of Erdo s and Graham by showing that for every positive integer r there are infinitely many positive integers n with the property that where p(1)=2, p(2)=3, p(3)=5, ... is the sequence of primes in ascending order, and e p (m) denotes the order of the prime p in t

On the Number of Sparse RSA Exponents
On the Number of Sparse RSA Exponents
✍ William D. Banks; Igor E. Shparlinski πŸ“‚ Article πŸ“… 2002 πŸ› Elsevier Science 🌐 English βš– 109 KB

An RSA modulus is a product M ¼ pl of two primes p and l. We show that for almost all RSA moduli M, the number of sparse exponents e (which allow for fast RSA encryption) with the property that gcdðe; jðMÞÞ ¼ 1 (hence RSA decryption can also be performed) is very close to the expected value.

On the parity of planar covers
On the parity of planar covers
✍ Dan Archdeacon; R. Bruce Richter πŸ“‚ Article πŸ“… 1990 πŸ› John Wiley and Sons 🌐 English βš– 263 KB

## Abstract A __covering__ is a graph map Ο•: __G__ β†’ __H__ that is an isomorphism when restricted to the star of any vertex of __G__. If __H__ is connected then |Ο•^βˆ’1^(__v__)| is constant. This constant is called the __fold number__. In this paper we prove that if __G__ is a planar graph that cover

On the decay exponent of isotropic turbu
On the decay exponent of isotropic turbulence
✍ M. Oberlack πŸ“‚ Article πŸ“… 2002 πŸ› John Wiley and Sons βš– 138 KB πŸ‘ 1 views

## On the decay exponent of isotropic turbulence It has long been observed that after a short initial transient period of time the decay of the velocity fluctuations u 2 of high Reynolds number homogeneous isotropic turbulence follows closely the algebraic law u 2 ∼ t -n . From experiments and DNS

On the Prime Power Factorization of n!
On the Prime Power Factorization of n!
✍ Yong-Gao Chen; Yao-Chen Zhu πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 105 KB

The parity of exponents in the prime power factorization of n! is considered. We extend and generalize Berend's result in [On the parity of exponents in the factorization of n!,