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
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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.
- Square numbers
- Triangular numbers
- Representation as difference of integers
2010 Mathematics Subject Classification
- 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.
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.