The Fibonacci sequence modulo p² -- An investigation by computer for p < 10¹⁴

2010/06/04 by Andreas-Stephan Elsenhans, Elsenhans, Andreas-Stephan, Jörg Jahnel +1
Computer Science · Mathematics · Physics and Astronomy · #11A41 #11B39 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11-04 #Secondary 11Y55 #math.NT #msc:11-04 #msc:11A41 #msc:11B39 #msc:11Y55

paper · pdf · doi:10.48550/arxiv.1006.0824

arxiv created 2010/06/04 · openalex publication_date 2010/06/04 · arxiv updated 2010/06/07 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We show that for primes p < 10¹⁴ the period length κ(p²) of the Fibonacci sequence modulo p² is never equal to its period length modulo p. The investigation involves an extensive search by computer. As an application, we establish the general formula κ(pⁿ) = κ(p) ⋅ p^n-1 for all primes less than 10¹⁴.

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