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

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.


Introduction
This paper brings together following, recently discovered, new aspects of Mertallic Ratios.
1) The Generalised Geometric Construction of all Metallic Ratios: cited by the Wikipedia in its page on "Metallic Mean" [6]. This generalised geometric substantiation of all Metallic Means was published in January 2021 [7] 2) The Mathematical Formula that provides the precise correlation between different Metallic Means. This explicit formula has been recently published in the month of May 2021 [1].
These couple of important aspects of Metallic Means: viz. the generalised geometric constructions of all Metallic Means and the concerned mathematical formulae, were brought together in the work mentioned in Reference [2].
The prime objective of current paper is to further synergize the these two newly discovered aspects of Metallic Means.
The synergism between above two features of Metallic Means unveils an intriguing pattern of Metallic Ratios, which asserts that the mathematical implications of these Means have not been fully appreciated so far. The abovementioned Geometry and Mathematics synergically enable us to recognize the full worth of these Metallic Means, as described in this paper.
As a brief introduction, each Metallic Mean n is the root of the simple Quadratic Equation X 2 -nX -1 = 0, where n is any positive natural number.
Thus, the fractional expression of the n th Metallic Ratio is n = + √ ² + Moreover, each Metallic Ratio can be expressed as the continued fraction: ; And hence, n = n + …..References: [3], [4], [5] GEOMETRIC CONSTRUCTION OF ALL METALLIC MEANS : The n th Metallic Mean ( n) = Hypotenuse + Cathetus Such Right Triangle not only provides for the accurate geometric construction and precise fractional expression of any n th Metallic Mean ( n), but its every geometric feature is the prototypical form of that Metallic Mean [6], [7], [8], [9].
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 Metallic Mean (φ 3 ).  Noticeably, n=6 forms such multiple triads: : Shaded Triads have been exemplified above.
And, just like n=6 exemplified above, every integer forms such multiple triads: Also noticeably, the Even Integers ( Even ns ) form comparatively more Triads than the Odd ns . Several such patterns about these Triads of Metallic Means have been discussed in detail in Reference [2]. Here, let us consider the classical correspondence of the abovementioned Formula and TRIADS with the Geometry of Metallic Ratios.
Remarkably, the abovementioned Triads of Metallic Means can be represented geometrically, as shown below.
For instance, the Triad [ n, m, k ] is illustrated geometrically in following Figure 2.  The following Table 1 shows associated Pythagorean Triples for first few values of n: Consider again the Figure 2 representing the Triad of Metallic Means [ n, m, k ]. 2θ1 = ( θ1 + θ2 + θ3 ) = The Smaller Acute Angle of associated Pythagorean Triple.

Special cases :
1) For n=1 : the right Triangle that represents Golden Ratio has its Cathetus 1 longer than its Second Cathetus  [7]. And second: the close correspondence between the Right Triangles representing Metallic Ratios and the Primitive Pythagorean Triples, as described here. These couple of facts clearly highlight the underlying proposition that the Metallic Means are more closely associated with; and more holistically represented by the "Right Angled Triangles", rather than Pentagon, Octagon or any other (n 2 +4)-gon.

Metallic Means and Pythagorean Primes : The Prime Families of Metallic Means
From the close correspondence between Metallic Means and Pythagorean Triples described so far, it becomes obvious that various Metallic Means are also closely associated with different Pythagorean Primes.
Consider the radical (n 2 +4) in the Fractional expression of the n th Metallic Mean ( n). By Fermat's Theorem on Sums of Two Squares, this radical is an integer multiple of a prime of the form p ≡ 1(mod 4). The Greatest Prime Factor ( i.e. the Largest Prime Divisor) of this radical (n 2 + 4) is a Pythagorean Prime, as shown below in Table 2. n n 2 +4 Greatest Prime Factor of (n 2 +4) : A Pythagorean Prime  It is noticeable from above table that various values of n have 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 13.
Hence, the different Metallic Means can be classified into various groups corresponding to the Greatest Prime Factors (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 (59) 2 + 4 = (41 × 17 × 5) (49) 2 + 4 = (37 × 13 × 5), and so on.
Noticeably, the Greatest Prime Factor of (n 2 +4) for n=43 is 109 (43) 2 + 4 = 109 × 17 However, why it has been included in the Prime 17 Family in above Table, that will become obvious with following couple of illustrations.
Remarkably, the Metallic Means belonging to same Prime Family exhibit very distinctive relations among themselves, as illustrated below. 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.
On the last note, it is worth mentioning here that several other intriguing properties of Metallic Means and their abovementioned TRIADS are described in details in the works mentioned in the References. For instance, these TRIADS of Metallic Means are found to be closely associated with the Pascal's Triangle [11]; the geometric substantiation of Metallic Ratios and their TRIADS [6] [7] [8] [9]; and special positions of Integers 3, 6 and 9 in the realm of Metallic Means [2] [12]. Further, all imperical formulae those provide the precise relations between different Metallic Means are described in the work mentioned in Reference [1].

Conclusion:
This paper brought together the generalised geometry of all Metallic Means and the mathematical formula that provides the precise correlations between different Metallic Ratios. The paper illustrated the concept of the "Triads of Metallic Means", and also the close correspondence between Metallic Ratios and the Pythagorean Triples as well as Pythagorean Primes.