An alternative proof of Nyblom’s results and a generalisation

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
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A. David Christopher
Department of Mathematics, The American College
Tamil Nadu, India

Abstract

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

    \[p(n,k)=\begin{cases}n(n+k), & \text{if}\ k\equiv 0\pmod 2;\\\dfrac{n(n+k)}{2}, & \text{if}\ k\equiv 1\pmod 2,\end{cases}\]

and D(n,k) to be the number of ways n can be expressed as a difference of two elements from the sequence p(n,k). Nyblom found closed expressions for D(n,0) and D(n,1) 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 D(n,k), an expression for D(n,k) is presented in terms of restricted form of D(n,0) and D(n,1). Also we consider another function due to Nyblom, denoted p_D(n), which counts the number of partitions of n with parts in arithmetic progression having common difference D. Nyblom and Evan found a simple expression for p_2(n) and put p_D(n) in terms of a divisor-counting functions when D\geq 3. Here we re-establish Nyblom’s expression for p_2(n), and find equinumerous expressions for p_D(n) when D\geq 3. Finally, we present the following generalised version of D(n,k): given a set of positive integers say, A, we denote by D(n,A), the number of ways n can be written as a difference of two elements from the set A. And we express D(n,A) in terms of partition enumerations when some restrictions are imposed upon the elements of A. We close with the hint that, boundedness of D(n,A) together with the divergence of \sum _{a\in A}\dfrac{1}{a} disproves Erdős arithmetic progression conjecture.

Keywords

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

2010 Mathematics Subject Classification

  • 11A67
  • 11B34

References

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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.

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