What does this research mean for the field?

The newly introduced Universal Metallic Family of quadratic irrationals, defined by α(n,N) = n + √(n²+N), unifies classical Metallic Means, Decimal-Metallic Ratios, and primitive Pythagorean triples under a single parameter N. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to introduce the Universal Metallic Family and demonstrate its connections to various mathematical concepts.

June 15, 2026Open Access

The Universal Metallic Family: Quadratic Irrationals n + √(n²+N), Base-N Shift Property, and Primitive Pythagorean Triples

Key Points

The aim is to introduce the Universal Metallic Family and demonstrate its connections to various mathematical concepts.
Defined quadratic irrationals α(n,N) = n + √(n²+N) for positive roots of x²−2nx−N=0.
Proved ten results uniformly for every positive integer N, including key identities and classifications.
Classified the family into two classes based on whether N is a perfect square.
Proved the Base-N Shift Theorem, applicable for every positive integer N.
Established the Universal Crown Identity and Universal Cassini Identity.
Demonstrated a complete Parity Classification Theorem for the Universal Metallic Family.

Abstract

This paper introduces the Universal Metallic Family, an infinite two-parameter family of quadratic irrationals α(n,N) = n + √(n²+N), defined as the positive roots of x²−2nx−N=0. The family simultaneously unifies and classifies the classical Metallic Means, the Deca-Metallic Ratios, the Decimal-Metallic Ratios, and every primitive Pythagorean triple as natural special cases governed by the single parameter N. Ten results are proved uniformly for every positive integer N, including the Base-N Shift Theorem, the Universal Crown Identity, the Universal Cassini Identity, a transcendental hyperbolic representation, and a complete Parity Classification Theorem (necessary and sufficient). The family admits a definitive two-class partition: Class A (perfect square N) and Class B (non-square N), with the Decimal-Metallic family identified as the canonical base-10 subfamily.

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