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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## Details

### Authors and affiliations

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, , and the silver ratio, , the integers approximated by their powers, namely the Lucas () and Pell-Lucas () numbers, never coincide except in trivial cases for which . The equation has only four solutions for , . The largest such encounter arises between and whilst the separation between larger pairings, and , 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

APASpivey, 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.

ChicagoRobin 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.

MLARobin 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.