Metallic Ratios, Pythagorean Triples & p≡1(mod 4) Primes : Metallic Means, Right Triangles and the Pythagoras Theorem

Authors

  • Chetansing Rajput Dr. , ACST

DOI:

https://doi.org/10.24297/jam.v20i.9075

Keywords:

Metallic Ratio, Pascal’s Triangle, Golden Ratio, Bronze Ratio, Pythagorean Triples, 3 6 9, Metallic Ratio Triads, Metallic Numbers, Right Triangle, Silver Ratio, Phi, Pi, Lucas, Pell, Fibonacci, Pythagoras Theorem, Metallic Mean

Abstract

This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper illustrates the concept of the “Triads of Metallic Means”, and aslo the close correspondence between Metallic Ratios and the Pythagorean Triples as well as Pythagorean Primes.

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References

Rajput, Chetansing (2021). Metallic Ratios : Beyond the Golden Ratio; The Mathematical Relationships between different Metallic Means. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 158-166. https://doi.org/10.24297/jam.v20i.9023

Rajput, Chetansing (2021). “Metallic Means : Beyond the Golden Ratio, New Mathematics and Geometry of all Metallic Ratios based upon Right Triangles, The Formation of the Triples of Metallic Means, And their Classical Correspondence with Pythagorean Triples and p≡1(mod 4) Primes, Also the Correlation between Metallic Numbers and the Digits 3 6 9”, JOURNAL OF ADVANCES IN MATHEMATICS, 20, 250–266. https://doi.org/10.24297/jam.v20i.9056

Vera W. de Spinadel (1999). The Family of Metallic Means, Vismath 1(3) from Mathematical Institute of Serbian Academy of Sciences and Arts.

Weisstein, Eric W. "Table of Silver means". MathWorld.

"An Introduction to Continued Fractions: The Silver Means", maths.surrey.ac.uk.

Rajput, Chetansing (2021). "A Right Angled Triangle for each Metallic Mean". Journal of Advances in Mathematics. 20: 32–33. https://en.wikipedia.org/wiki/Metallic_mean#cite_note-15

Rajput, Chetansing (2021). Golden Ratio. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 19–42. https://doi.org/10.24297/jam.v20i.8945

Rajput, Chetansing (2021). Metallic Means and Right Triangles: The Geometric Substantiation of all Metallic Ratios JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9029

Rajput, Chetansing (2021). Golden Ratio and other Metallic Means: The Geometric Substantiation of all Metallic Ratios with Right Triangles. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 174-187. https://doi.org/10.24297/jam.v20i.9034

Rajput, Chetansing (2021). Metallic Ratio Triads, The Mathematical and Geometric Relations between different Metallic Means, Metallic Numbers and the Right Angled Triangles. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. ttps://doi.org/10.24297/jam.v20i.9029

Rajput, Chetansing (2021). Metallic Ratios and Pascal’s Triangle : Triads of Metallic Means in the Pascal’s Triangle. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9029

Rajput, Chetansing (2021). Metallic Ratios and the Digits 3 6 9, Mathematical Relations between different Metallic Means, And the Special Significance of Digits 3, 6, 9 in the realm of Metallic Numbers. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 167-173. https://doi.org/10.24297/jam.v20i.9029

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Published

2021-07-02

How to Cite

Rajput, C. (2021). Metallic Ratios, Pythagorean Triples & p≡1(mod 4) Primes : Metallic Means, Right Triangles and the Pythagoras Theorem. JOURNAL OF ADVANCES IN MATHEMATICS, 20, 267–282. https://doi.org/10.24297/jam.v20i.9075

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