Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles

The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.

where Fn, Fn+1…… is the Fibonacci-like Integer Sequence (Fibonacci Sequence for Golden Ratio, Pell Sequence for Silver Ratio, Bronze-Fibonacci Sequence for Bronze Ratio, etc.) and Ln, Ln+1…….. is the corresponding Lucas Sequence associated with that particular Metallic Mean.
The following Table 1 shows associated Pythagorean Triples for first few Metallic Means: Noticeably, certain patterns can be observed from above table.
Consider the radical (n 2 + 4) in the fractional expression of the n th Metallic Mean : n= n + √ ² + 2 The n th Metallic Mean n is epitomized by such Primitive Pythagorean Triple whose Hypotenuse is Factor of this Radical (n 2 + 4); with following sub-rules : ) From the Fifth Metallic Ratio onwards, any n th Metallic Mean n is given by the generalized formula: Another important reason for not having any Pythagorean Triple that may represent δ2 would become obvious in the subsequent part of this paper. Noteworthy here, this Silver Ratio forms "TRIADS" with 1 , 4 and 6 (the Metallic Means having common associated Pythagorean Triple 3-4-5); and also with 3 and 10 (the Metallic Means having common associated Pythagorean Triple 5-12-13). The concept of these "TRIADS of Metallic Means" will be elaborated in the subsequent part of this paper.
Right Triangles are more "Metallic" than any 'n 2 +4'gon  , and [4] to [7]. The characteristic geometry of such Right Triangle having its catheti 1 and , is resplendent with the corresponding n th Metallic Mean ( n) embedded in its every geometric aspect.
For example, the remarkable expression of Golden Ratio in every geometric feature of 1:2:√5 triangle, including all its angles and side lengths, its 'Incenter-Excenters Orthocentric system', its Gergonne and Nagel triangles, and also the Nobbs points and the Gergonne line, various triangle centers as well as the Incircle of 1:2:√5 triangle, make this triangle the quintessential form of the Golden Ratio (φ) and also of the fourth Moreover, such Fractional Expression Triangle is also the Limiting Triangle for the Pythagorean Triples formed with the Hypotenuses those equal the alternate terms of the Integer Sequence associated with that Metallic Mean ( n). For example, the Pythagorean triples derived from Fibonacci series, approach the 1:2:√5 triangle's proportions, as the series advances. Likewise, the Pythagorean triples having alternate Pell Numbers as their Hypotenuses, approach the 1:1:√2 triangle's proportions, as the series advances, and so on.

Metallic Ratios and various Right Triangles :
Remarkably, all Metallic Ratios can be precisely expressed with various Right Angled Triangles. Three of the more intriguing such triangles are :  Several properties of these Triads are described in the works mentioned in References, like presence of these      It is noticeable from above table that multiple values of n exhibit common Greatest Prime Factor of (n 2 + 4).
For example, for 3 th , 10 th and 16 th Metallic Means, the common Greatest Prime Factor of the radical (n 2 + 4) is

Hence, the different Metallic Means can be classified into various groups corresponding to the Greatest Prime Factor (GPF) of the radical (n 2 +4).
This GPF is necessarily a Pythagorean Prime (4x + 1), as shown below in Table 3. Noticeably, as described in previous section: the Hypotenuse of associated Pythagorean Triple is a factor of (n 2 +4), and the associated Pythagorean Primes, as shown in Table 3 are the Greatest Prime Factors of (n 2 +4).
Note: the 8 th and the 9 th Metallic Means both have Pythagorean Prime 17 as the GPF of their (n 2 +4), however they have different associated Primitive Pythagorean Triples, as shown in Table 1 Likewise, consider another example for illustration and comparison. the Metallic Means associated with the Pythagorean Prime 17 : 8 , 9 and 26 Such several distinctive correlations are observed among the Metallic Means belonging to the same Pythagorean Prime Families, and these correlations are bound to generate more such intriguing mathematical formulae, which may provide the precise relations between different Metallic Ratios.

3, 6 and 9 in the Realm of Metallic Means : Triangles, Triads, Triples, and now 3, 6, 9 !
The proponents of Vortex Based Mathematics will continue to make irrational claims, and their opponents will continue to debunk them on grounds of the Base-10 Number System. Let the both camps do their jobs with missionary zeal ! Author's objective is just to appreciate the beauty of numbers and the special attributes of the digits 3, 6 and 9, especially their unique patterns in the realm of Metallic Means.
Consider the Triads of Metallic Means [ n, m, k ] with various integer values of n, shown below in Table 4.
Noticeably, in the following Table 4 : If n is NOT multiple of 3, the alternate values of m and k have their digital roots 3, 6, or 9.
And, if n is multiple of 3 : None of the associated ms and ks have their digital roots 3, 6, or 9.
Hence, only one of the n, m and k values in a Triad can have the Digital Root 3, 6 or 9. : Alternate ms and ks have their digital roots 3, 6, or 9.
For n= 3, 6, 9 : None of the associated ms and ks have their digital roots 3, 6, or 9. More remarkably, the number of Triads formed ( or the numbers of ms and ks ) increase noticeably for n = 6 and 9 For Even ns : the number of Triads exhibit noticeable rise at n = 6, 16, 26, 36…….and so on.
For Odd ns : the number of Triads exhibit noticeable rise at n = 9, 19, 29…….and so on.
Moreover, it can be noticed from above Table : if n is multiple of 3, the Digital Root of | − | is 3, 6 or 9.
Consider the Triads of Metallic Means formed with n = 6, and the values of (m-n), as shown below.
Note the bottom row in above table which contains the values of (m-n).
For Odd ns : the pattern based upon product of the Prime Factors of ( n 2 + 4 ) is observed.
In either case, the integers 3, 6 and 9 are conspicuous by their absence from these (m-n) or (k-n) values.
Noticeably, the integer 7 is present not directly as (m-n), but it's present only as the Digital Roots of certain (m-n) values; for instance the red shaded number 232 in above example. Such presence of 7 as Digital Root of (m-n) values is observed with n = 8,10,11,14,16,22,26,29,34,36,39, and so on. Remarkably, with n = 26, 36, 39, etc. multiple (m-n) values are found to have their digital root Seven.
However, the integers 3, 6 and 9 are invariably missing from this (m-n) pattern, they are neither present directly as (m-n) value, nor as the digital root of any (m-n) or (k-n) values.
As mentioned above, if n is multiple of 3, the Digital Root of |k − m| is 3, 6 or 9. And, if n is NOT multiple of 3, the Digital Roots of the |k − m| value are 1,2,4,5,7,8……..
Further, the Triads of Metallic Ratios with n = 9 and the multiples of nine 9 exhibit their own characteristic patterns.
The point is that the Numbers 3, 6 and 9 exhibit their very peculiar and distinctive attributes, in the dominion of Metallic Numbers.
Moreover, the idiosyncrasy of 3, 6 and 9 is exhibited in several other such patterns of Metallic Means and their Triads.