LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The Pell norm is multiplicative

Statement

Let $α = x + y \sqrt{D}, β = u + v \sqrt{D} \in Z [\sqrt{D}] .$ Then $α β = (x u + D y v) + (x v + y u) \sqrt{D},$ and the Pell norm is multiplicative: $N_{D} (α β) = N_{D} (α) N_{D} (β) .$

Facts & Assumptions

Given: Two elements $α = x + y \sqrt{D}$ and $β = u + v \sqrt{D}$ of $Z [\sqrt{D}]$ .

[F1]

In $Z [\sqrt{D}]$ , one has

(x + y \sqrt{D}) (u + v \sqrt{D}) = (x u + D y v) + (x v + y u) \sqrt{D}

and $N_{D} (x + y \sqrt{D}) = x^{2} - D y^{2} .$ (The norm on the explicit order $Z [\sqrt{D}]$ ).

Proof

technique · direct

1.1

Fact [F1] gives the coordinate product formula immediately: $α β = (x u + D y v) + (x v + y u) \sqrt{D} .$

2.1

Applying the norm formula from [F1] to step 1.1 and expanding gives $N_{D} (α β) = (x u + D y v)^{2} - D (x v + y u)^{2} = (x^{2} - D y^{2}) (u^{2} - D v^{2}) = N_{D} (α) N_{D} (β) .$

F1step 1.1algebra∎

Depends on

The norm on the explicit order $\mathbb{Z}[\sqrt{D}]$

Used by

One generalized Pell solution generates infinitely many more Corollary
Pell-equivalence for generalized solutions Definition
Integral Pell solutions form an abelian group Proposition
Every Pell equation has a positive nontrivial integral solution Theorem
Generalized Pell solutions fall into finitely many Pell orbits Theorem

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Sources

Keith Conrad, Pell's Equation, I (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)