Balancing sequence contains no prime number

Shekh Mohammed Zahid
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 25–28
Full paper (PDF, 54 Kb)

Details

Authors and affiliations

Shekh Mohammed Zahid
Institute of Mathematics and Applications
Bhubaneswar, Odisha, India

Abstract

The study of prime number in any number sequence is crucial part. In recent year Panda and Behera introduced a new number sequence that is solutions of Diophantine equation 1 + 2 + 3 + … + (n – 1) = (n + 1) + (n + 2) + … + (n + r), where n and r are positive integers. The pairs (n, r) constitute a solution of above equation then n is called balancing number and r is the corresponding balancer. In this paper, we prove a main result that there is no prime number in the sequence of balancing numbers.

Keywords

• Diophantine equation
• Balancing numbers
• Prime numbers
• Cobalancing numbers

AMS Classification

• 11B83
• 11A41

References

1. Behera, A., & Panda, G. K. (1999) On the square roots of triangular numbers, Fibonacci Quarterly, 37, 98–105.
2. Ray, P. K. (2009) Balancing and Cobalancing Numbers, http://www.nitrkl.ac.in/, http://ethesis.nitrkl.ac.in/2750/.
3. Panda, G. K., & Ray, P. K. (2005) Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 8, 1189–1200.
4. Panda, G. K., and Ray, P. K. (2011) Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 6, 41–72.