Close encounters of the golden and silver ratios

Robin James Spivey
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 170-184
DOI: 10.7546/nntdm.2019.25.3.170-184
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Robin James Spivey
GeoVista, 10 Glan Y Môr, Glan Conwy, LL28 5SP, United Kingdom

Abstract

What are the nearest approaches of the natural powers of two irrational numbers, allowing for arbitrarily large exponents? In the case of the first two metallic means, a definitive answer to this challenging question lies within reach. Despite the small magnitude of the golden ratio, \phi=(1+\sqrt{5})/2\approx1.618, and the silver ratio, \delta_s=1+\sqrt{2}\approx2.414, the integers approximated by their powers, namely the Lucas (L_m\approx\phi^m) and Pell-Lucas (U_n\approx\delta_s^n) numbers, never coincide except in trivial cases for which m=0. The equation L_m=U_n\pm1 has only four solutions for m>0, n>0. The largest such encounter arises between L_{11}=199 and U_6=198 whilst the separation between larger pairings, m>11 and n>6, always exceeds 42.

Keywords

  • Lucas series
  • Pell–Lucas series
  • Golden ratio
  • Silver ratio
  • Metallic means.

2010 Mathematics Subject Classification

  • 11B39
  • 14G05
  • 11D25

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Cite this paper

APA

Spivey, R. J. (2019). Close encounters of the golden and silver ratios. Notes on Number Theory and Discrete Mathematics, 25(3), 170-184, doi: 10.7546/nntdm.2019.25.3.170-184.

Chicago

Robin James Spivey . (2019). “Close encounters of the golden and silver ratios.” Notes on Number Theory and Discrete Mathematics. Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 170-184, doi: 10.7546/nntdm.2019.25.3.170-184.

MLA

Robin James Spivey. (2019). “Close encounters of the golden and silver ratios” Notes on Number Theory and Discrete Mathematics 25.3 (2019): 170-184. Print, doi: 10.7546/nntdm.2019.25.3.170-184.

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