Document Type
Article
Publication Date
7-16-2020
Publication Title
Journal of Number Theory
Volume
221
First Page
222
Last Page
231
DOI
10.1016/j.jnt.2020.06.002
Keywords
Almost prime, Small gaps, Erdos, Mirsky, Divisor, Exponent pattern
Disciplines
Mathematics | Number Theory
Abstract
We show that for any positive integer n, there is some fixed A such that d(x) = d(x +n) = A infinitely often where d(x) denotes the number of divisors of x. In fact, we establish the stronger result that both x and x +n have the same fixed exponent pattern for infinitely many x. Here the exponent pattern of an integer x > 1is the multiset of nonzero exponents which appear in the prime factorization of x.
Recommended Citation
Daniel A. Goldston, Sidney W. Graham, Apoorva Panidapu, Janos Pintz, Jordan Schettler, and Cem Y. Yıldırım. "Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers II" Journal of Number Theory (2020): 222-231. https://doi.org/10.1016/j.jnt.2020.06.002
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
