Brahmagupta-Fibonacci Identity

Summary: The multiplicative identity $(p^2+c{q}^2)(r^2+c{s}^2)=x^2+c{y}^2$ where $x=pr-cqs$, $y=ps+qr$ (with a second representation by sign-flipping $q$). Shows that the set of integers representable as $x^2+c{y}^2$ is closed under multiplication. For $c=1$ this is the classical sum-of-two-squares identity; for $c=-N$ it gives Pell composition. Euler derives it in ch2.0.11-quadratic-form-factorization §170 by complex factorization $x^2+c{y}^2=(x+y\sqrt{-c})(x-y\sqrt{-c})$.

Sources: chapter-2.0.11, chapter-2.0.12, additions-9

Last updated: 2026-05-10

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Statement

For any rationals $p,q,r,s$ and any constant $c$:

$\boxed{(p^2+c{q}^2)(r^2+c{s}^2)=(pr-cqs{)}^2+c(ps+qr{)}^2.}$

A second representation (since only $q^2$ appears in the LHS):

$(p^2+c{q}^2)(r^2+c{s}^2)=(pr+cqs{)}^2+c(ps-qr{)}^2.$

So when both representations give distinct $(x,y)$, the product has two $x^2+c{y}^2$ representations.

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Special Cases

$c=1$: Sum of Two Squares (Diophantus, Brahmagupta, Fibonacci)

$(p^2+q^2)(r^2+s^2)=(pr-qs{)}^2+(ps+qr{)}^2=(pr+qs{)}^2+(ps-qr{)}^2.$

This is the Fibonacci identity (1225, in Liber Quadratorum) and was known earlier to Diophantus (3rd century) and Brahmagupta (7th century).

Worked example: $(3^2+2^2)(2^2+1^2)=13\cdot 5=65=4^2+7^2=8^2+1^2$.

$c=-N$: Pell Composition

$(p^2-N{q}^2)(r^2-N{s}^2)=(pr+Nqs{)}^2-N(ps+qr{)}^2.$

This is Brahmagupta’s bhāvanā (composition law) for Pell’s equation: solutions of $x^2-N{y}^2=1$ form a multiplicative semigroup. See pell-equation and ch2.0.7-pell-equation-method.

$c=2$: Sum of Square and Twice a Square

$(p^2+2q^2)(r^2+2s^2)=(pr-2qs{)}^2+2(ps+qr{)}^2.$

Used in ch2.0.11-quadratic-form-factorization §173 to characterize numbers of form $x^2+2y^2$.

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Derivation via Complex Factorization

Factor $x^2+c{y}^2=(x+y\sqrt{-c})(x-y\sqrt{-c})$. If

$x+y\sqrt{-c}=(p+q\sqrt{-c})(r+s\sqrt{-c}),$

then by conjugation $x-y\sqrt{-c}=(p-q\sqrt{-c})(r-s\sqrt{-c})$, so

$x^2+c{y}^2=(p^2+c{q}^2)(r^2+c{s}^2).$

Expanding the RHS of the first equation:

$(p+q\sqrt{-c})(r+s\sqrt{-c})=pr-cqs+(ps+qr)\sqrt{-c},$

reading rational and irrational parts gives $x=pr-cqs$, $y=ps+qr$.

This is multiplication in $\mathbb{Z}[\sqrt{-c}]$ followed by the norm map: $N(\alpha )N(\beta )=N(\alpha \beta )$ where $N(p+q\sqrt{-c})=p^2+c{q}^2$.

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Lagrange’s Generalisation (Add. IX, Art. 89)

The identity is the quadratic case of a general construction of norm-forms: for any monic polynomial $s^2-as+b=0$ with roots $\alpha ,\beta$, $(x+\alpha y)(x+\beta y)=x^2+axy+b{y}^2$ and the composition law $(x^2+axy+b{y}^2)(x^{\prime 2}+a{x}^{\prime }{y}^{\prime }+b{y}^{\prime 2})=X^2+aXY+b{Y}^2$ holds with $X=x{x}^{\prime }-by{y}^{\prime }$ and $Y=x{y}^{\prime }+y{x}^{\prime }+ay{y}^{\prime }$. Setting $a=0$ recovers the identity above (with $b=c$). Setting $a=0,b=-A$ gives the Pell composition.

Lagrange extends this to cubic and higher norm forms in add9-norm-forms-composition, anticipating Gauss’s composition of binary quadratic forms and the general theory of norms in algebraic number fields.

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Generalized Form: Coefficient on First Term

For $a{x}^2+c{y}^2$ (with $a\ne 1$), the analogous identity (Euler §177–179) is

$\boxed{(a{p}^2+c{q}^2)(r^2+ac{s}^2)=(pr-cqs{)}^{2}a+c(qr+aps{)}^2\cdot a^{-1}\cdot ?}$

More cleanly, with $x=pr-cqs$, $y=qr+aps$:

$a{x}^2+c{y}^2=(a{p}^2+c{q}^2)(r^2+ac{s}^2),$

mixing two distinct quadratic forms (Type I and Type II in §177’s terminology). See ch2.0.11-quadratic-form-factorization for the proto-genus theory.

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Higher Powers

The same factor-of-conjugate trick at higher powers gives parametrizations for $x^2+c{y}^2$ being a cube, biquadrate, etc., by raising $p+q\sqrt{-c}$ to the desired exponent:

$(p+q\sqrt{-c}{)}^n=X+Y\sqrt{-c}⟹X^2+c{Y}^2=(p^2+c{q}^2{)}^n.$

This is the basis of ch2.0.12-quadratic-form-as-power.

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Applications

• Sums of two squares (sums-of-two-squares): closure under multiplication; classification of representable integers.
• Pell composition (pell-equation): generating infinite solutions from a fundamental one.
• Norms in number rings: foundation for ring-theoretic arithmetic in $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{-2}]$, etc.
• Rationalization (rationalization): related to multiplying by conjugates.

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Historical Note

Mathematician	Date	Form known
Diophantus	c. 250	$c=1$, geometric form
Brahmagupta	628	All $c$, Brāhmasphuṭasiddhānta (composition law)
Fibonacci	1225	$c=1$, Liber Quadratorum
Euler	1770	All $c$ via complex factorization, Elements of Algebra

The modern name “Brahmagupta-Fibonacci identity” typically refers to $c=1$; the “Brahmagupta identity” to general $c$.

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Related pages

• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• sums-of-two-squares
• pell-equation
• imaginary-numbers
• rationalization
• algebraic-identities
• norm-forms
• composition-of-forms
• binary-quadratic-forms
• add9-norm-forms-composition