On the Divisibility a! + b! | (a + b)!

Abstract

In this article, we investigate the pairs of positive integers for which sum of their factorials divides the factorial of their sum and establish a bound on their difference. We also solve the divisibility question over the set of Fibonacci numbers. We conclude by proving that there are infinitely many such pairs of positive integers with difference 2 and conjecture that for any positive integer k there are infinitely many such pairs (a, b) with $|a-b|=k.$

MSC::

• Primary 11B65
• Secondary 11B39
• 11C08

Acknowledgment

The author thanks the anonymous referees for their helpful suggestions and comments.

Additional information

Notes on contributors

Ayan Nath

Ayan Nath is a high school student at Kaliabor College, Assam, India. He has qualified for the Indian National Mathematical Olympiad (INMO) and has attended, multiple times, the Indian IMO Training Camp (IMOTC), which is an equivalent of the MOP (Mathematical Olympiad Program, organized in the USA). He is active on the Art of Problem Solving website under the username ayan.nmath.