A. David Christopher

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 2, Pages 116–126

DOI: 10.7546/nntdm.2020.26.2.116-126

**Full paper (PDF, 191 Kb)**

## Details

### Authors and affiliations

A. David Christopher

*Department of Mathematics, The American College
Tamil Nadu, India
*

### Abstract

Let be a positive integer and be a non-negative integer. We define

and to be the number of ways can be expressed as a difference of two elements from the sequence . Nyblom found closed expressions for and in terms of some restricted number-of-divisors functions. Here we re-establish these two results of Nyblom in a relatively simple way. Along with the other interpretations for , an expression for is presented in terms of restricted form of and . Also we consider another function due to Nyblom, denoted , which counts the number of partitions of with parts in arithmetic progression having common difference . Nyblom and Evan found a simple expression for and put in terms of a divisor-counting functions when . Here we re-establish Nyblom’s expression for , and find equinumerous expressions for when . Finally, we present the following generalised version of : given a set of positive integers say, , we denote by , the number of ways can be written as a difference of two elements from the set . And we express in terms of partition enumerations when some restrictions are imposed upon the elements of . We close with the hint that, boundedness of together with the divergence of disproves Erdős arithmetic progression conjecture.

### Keywords

- Square numbers
- Triangular numbers
- Representation as difference of integers

### 2010 Mathematics Subject Classification

- 11A67
- 11B34

### References

- Cook, R., & Sharp, D. (1995). Sums of arithmetic progressions, Fibonacci Quart., 33, 218–221.
- Christopher, A. D. (2015). Partitions with Fixed Number of Sizes, J. Integer Seq., 18, Article 15.11.5.
- Mason, T. E. (1912). On the representation of an integer as the sum of consecutive integers, Amer. Math. Monthly, 19 (3), 46–50.
- Munagi, A. O. (2010). Combinatorics of integer partitions in arithmetic progression, Integers, 10, 73–82.
- Nyblom, M. A. (2001). On the Representation of the Integers as a Difference of

non-consecutive Triangular numbers, Fibonacci Quart., 39 (3), 256–263. - Nyblom, M. A. (2002). On the Representation of the Integers as a Difference of Squares, Fibonacci Quart., 40 (3), 243–246.
- Nyblom, M. A., & Evans, C. (2003). On the enumeration of partitions with summands in arithmetic progression, Australas. J. Combin., 28, 149–159.

## Related papers

## Cite this paper

Christopher, A. D. (2020). An alternative proof of Nyblom’s results and a generalisation. *Notes on Number Theory and Discrete Mathematics*, 26 (2), 116-126, DOI: 10.7546/nntdm.2020.26.2.116-126.