Some Fibonacci congruences with square moduli

Anthony G. Shannon, Peter J.-S. Shiue and Christopher Saito
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 229–241
DOI: 10.7546/nntdm.2026.32.1.229-241
Full paper (PDF, 715 Kb)

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Authors and affiliations

Anthony G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW 2033, Australia

Peter J.-S. Shiue
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, Nevada, 89154-4020, USA

Christopher Saito
Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, Nevada, 89154-4020, USA

Abstract

Fibonacci congruence with prime moduli have been extensively studied. Square moduli are obviously not prime numbers, so why study such congruences? The answer is to investigate patterns for a special class of moduli, since there are many Fibonacci identities containing squares of Fibonacci numbers. Previous studies do not seem to have revealed anything of note that is notably distinct from other composite moduli.

Keywords

• Basis vectors
• Fibonacci numbers
• Integral lattices
• Jacobsthal sequences
• Pythagorean triples
• Sophie Germain

2020 Mathematics Subject Classification

• 11B39
• 11B37
• 11D09
• 11H06

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Manuscript history

• Received: 17 January 2026
• Revised: 19 March 2026
• Accepted: 20 March 2026
• Online First: 21 March 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).