What do the Golden Ratio, the Silver Ratio, the Bronze Ratio, primitive Pythagorean triples, and the decimal number system have in common? This paper shows they are all faces of a single, infinite two-parameter family of numbers: α(n,N) = n + √(n² + N). By letting N range freely over the positive integers — not just powers of ten — this family unifies and classifies the classical Metallic Means of de Spinadel, the author's previously-introduced Deca-Metallic and Decimal-Metallic Ratios, and every primitive Pythagorean triple, all within one coherent algebraic, geometric, and hyperbolic framework. Two central theorems anchor the paper. The Base-N Shift Theorem shows that when N is a power of a base b, the fractional digits of α(n,N) reappear after an exact digit shift in that base — decimal shift for base 10, binary for base 2, and so on for any base. The Universal Crown Identity derives every member of the family directly from a primitive Pythagorean triple via a single geometric formula. A complete Classification Theorem partitions all positive integers N into two classes — perfect squares and non-squares — governed by a newly-proved Parity Classification Theorem describing exactly which Metallic Means appear, and how, at each layer of any base-b hierarchy. Along the way, the paper uncovers striking patterns: the Golden Ratio reappears scaled by 10, 100, 1000, 10000, ... at every even power of ten, and at N=100 alone the entire Metallic Means family — Golden, Silver, Bronze, and beyond — emerges scaled uniformly by ten. The family also generates generalized Fibonacci and Lucas sequences satisfying Binet formulas and a universal Cassini identity, connecting this work to the broader theory of integer sequences.
Chetansing Rajput (Mon,) studied this question.