Equalities between greatest common divisors involving three coprime pairs

Rogelio Tomás García
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 5–7
DOI: 10.7546/nntdm.2020.26.3.5-7
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Authors and affiliations

Rogelio Tomás García
CERN
Geneva, Switzerland

Abstract

A new equality of the greatest common divisor (\gcd) of quantities involving three coprime pairs is proven in this note. For a_i and b_i positive integers such that \gcd(a_i, b_i) = 1 for i \in \{ 1, 2, 3 \} and d_{ij} = |a_i b_j - a_j b_i|, then \gcd(d_{32},d_{31}) = \gcd(d_{32},d_{21}) = \gcd(d_{31}, d_{21}). The proof uses properties of Farey sequences.

Keywords

  • Greatest common divisor
  • Farey
  • Equality

2010 Mathematics Subject Classification

  • 11A05
  • 11B57

References

  1. Hardy, G. H., & Wright, E. M. (1996). An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications.

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Cite this paper

Tomás García, R. (2020). Equalities between greatest common divisors involving three coprime pairs. Notes on Number Theory and Discrete Mathematics, 26 (3), 5-7, DOI: 10.7546/nntdm.2020.26.3.5-7.

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