Two theorems on square numbers

Nguyen Xuan Tho
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 75–80
DOI: 10.7546/nntdm.2022.28.1.75-80
Download PDF (169 Kb)


Authors and affiliations

Nguyen Xuan Tho  
Hanoi University of Science and Technology
Hanoi, Vietnam


We show that if a is a positive integer such that for each positive integer n, a+n^2 can be expressed x^2+y^2, where x,y\in \mathbb{Z}, then a is a square number. A similar theorem also holds if a+n^2 and x^2+y^2 are replaced by a+2n^2 and x^2+2y^2, respectively.


  • Elementary number theory
  • Square numbers
  • Quadratic reciprocity

2020 Mathematics Subject Classification

  • 11A15
  • 11E04


  1. Ireland, K., & Rosen, M. (1998). A Classical Introduction to Number Theory (2nd ed.). Springer.
  2. Wang, S. (1950). On Grunwald’s theorem, Annals of Mathematics, Second Series, 51(2), 471–484.

Manuscript history

  • Received: 2 February 2021
  • Revised: 16 November 2021
  • Accepted: 11 February 2022
  • Online First: 14 February 2022

Related papers

Cite this paper

Tho, N. X. (2022). Two theorems on square numbers. Notes on Number Theory and Discrete Mathematics, 28(1), 75-80, DOI: 10.7546/nntdm.2022.28.1.75-80.

Comments are closed.