sq_add_sq_mul_sq_add_sq
theorem sq_add_sq_mul_sq_add_sq :
∀ {R : Type u_1} [inst : CommRing R] {x₁ x₂ y₁ y₂ : R},
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 + y₂ ^ 2) = (x₁ * y₁ - x₂ * y₂) ^ 2 + (x₁ * y₂ + x₂ * y₁) ^ 2
This theorem states the Brahmagupta-Fibonacci identity (or Diophantus identity), which expresses the product of sums of squares in a particular way. Specifically, for any type `R` that forms a commutative ring (denoted `[inst : CommRing R]`), and for any elements `x₁, x₂, y₁, y₂` of `R`, the product of the sum of squares of `x₁` and `x₂` with the sum of squares of `y₁` and `y₂` equals the sum of squares of `(x₁ * y₁ - x₂ * y₂)` and `(x₁ * y₂ + x₂ * y₁)`. In simple terms, it's analogous to multiplying two complex numbers.
More concisely: For any commutative ring `R` and elements `x₁, x₂, y₁, y₂` in `R`, the sum of the products of the squares of `x₁, x₂, y₁, y₂` equals the sum of the squares of `(x₁*y₁ − x₂*y₂)` and `(x₁*y₂ + x₂*y₁)`.
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pow_four_add_four_mul_pow_four
theorem pow_four_add_four_mul_pow_four :
∀ {R : Type u_1} [inst : CommRing R] {a b : R}, a ^ 4 + 4 * b ^ 4 = ((a - b) ^ 2 + b ^ 2) * ((a + b) ^ 2 + b ^ 2)
This is a statement of Sophie Germain's identity in the context of commutative rings. For any commutative ring `R` and elements `a` and `b` in `R`, the sum of the fourth power of `a` and four times the fourth power of `b` is equal to the product of the expression `((a - b) ^ 2 + b ^ 2)` and `((a + b) ^ 2 + b ^ 2)`. This identity is a useful algebraic identity with applications in number theory and is named after the French mathematician Sophie Germain.
More concisely: For any commutative ring `R` and elements `a, b` in `R`, `(a^4 + 4*b^4) = ((a-b)^2*(a+b)^2)`.
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pow_four_add_four_mul_pow_four'
theorem pow_four_add_four_mul_pow_four' :
∀ {R : Type u_1} [inst : CommRing R] {a b : R},
a ^ 4 + 4 * b ^ 4 = (a ^ 2 - 2 * a * b + 2 * b ^ 2) * (a ^ 2 + 2 * a * b + 2 * b ^ 2)
This is the theorem of Sophie Germain's identity in the field of mathematics. It states that for any commutative ring 'R' and for any elements 'a' and 'b' of the ring, the fourth power of 'a' plus four times the fourth power of 'b' is equal to the product of '(a ^ 2 - 2 * a * b + 2 * b ^ 2)' and '(a ^ 2 + 2 * a * b + 2 * b ^ 2)'. This identity holds true for all commutative rings and can be utilized in various mathematical proofs and computations.
More concisely: For any commutative ring 'R' and elements 'a' and 'b' in 'R', $(a^4 + 4b^4) = (a^2 \pm 2ab + 2b^2)^2$.
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sq_add_mul_sq_mul_sq_add_mul_sq
theorem sq_add_mul_sq_mul_sq_add_mul_sq :
∀ {R : Type u_1} [inst : CommRing R] {x₁ x₂ y₁ y₂ n : R},
(x₁ ^ 2 + n * x₂ ^ 2) * (y₁ ^ 2 + n * y₂ ^ 2) = (x₁ * y₁ - n * x₂ * y₂) ^ 2 + n * (x₁ * y₂ + x₂ * y₁) ^ 2
The theorem `sq_add_mul_sq_mul_sq_add_mul_sq` is a statement of Brahmagupta's identity in the context of a commutative ring. Given any elements `x₁`, `x₂`, `y₁`, `y₂`, and `n` of a commutative ring `R`, the square of `x₁` plus the product of `n` and the square of `x₂`, times the square of `y₁` plus the product of `n` and the square of `y₂`, equals the square of the difference between the product of `x₁` and `y₁` and the product of `n`, `x₂`, and `y₂`, plus `n` times the square of the sum of the product of `x₁` and `y₂` and the product of `x₂` and `y₁`. This identity is a fundamental property in number theory and is used in various proofs and constructions.
More concisely: In a commutative ring, the sum of the squares of `(x₁ + n*x₂)` and `(y₁ + n*y₂)`, is equal to the square of `(x₁*y₂ + x₂*y₁ - n*x₁*x₂ - n*y₁*y₂)` plus `n` times the sum of the squares of `(x₁*y₂)` and `(x₂*y₁)`.
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sum_four_sq_mul_sum_four_sq
theorem sum_four_sq_mul_sum_four_sq :
∀ {R : Type u_1} [inst : CommRing R] {x₁ x₂ x₃ x₄ y₁ y₂ y₃ y₄ : R},
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ^ 2
This theorem, known as Euler's four-square identity, states that for any commutative ring `R` and any eight elements `x₁`, `x₂`, `x₃`, `x₄`, `y₁`, `y₂`, `y₃`, `y₄` of `R`, the sum of the squares of the `x` elements multiplied by the sum of the squares of the `y` elements equals the sum of the squares of four different combinations of the `x` and `y` elements. These combinations mimic the multiplication of two quaternions. The identity is a fundamental result in number theory and has applications in the representation of numbers as sums of squares.
More concisely: For any commutative ring `R` and elements `x₁, x₂, x₃, x₄, y₁, y₂, y₃, y₄` in `R`, the quadratic forms `∑xi²` and `∑ji²` are equal if and only if there exist unique i, j in {1, 2, 3, 4} such that `x₁x₂ + x₃x₄ = yi² + yj²` and `x₁x₃ + x₂x₄ = yi yj`.
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sum_eight_sq_mul_sum_eight_sq
theorem sum_eight_sq_mul_sum_eight_sq :
∀ {R : Type u_1} [inst : CommRing R] {x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ : R},
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2 + x₅ ^ 2 + x₆ ^ 2 + x₇ ^ 2 + x₈ ^ 2) *
(y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2 + y₅ ^ 2 + y₆ ^ 2 + y₇ ^ 2 + y₈ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄ - x₅ * y₅ - x₆ * y₆ - x₇ * y₇ - x₈ * y₈) ^ 2 +
(x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃ + x₅ * y₆ - x₆ * y₅ - x₇ * y₈ + x₈ * y₇) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂ + x₅ * y₇ + x₆ * y₈ - x₇ * y₅ - x₈ * y₆) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁ + x₅ * y₈ - x₆ * y₇ + x₇ * y₆ - x₈ * y₅) ^ 2 +
(x₁ * y₅ - x₂ * y₆ - x₃ * y₇ - x₄ * y₈ + x₅ * y₁ + x₆ * y₂ + x₇ * y₃ + x₈ * y₄) ^ 2 +
(x₁ * y₆ + x₂ * y₅ - x₃ * y₈ + x₄ * y₇ - x₅ * y₂ + x₆ * y₁ - x₇ * y₄ + x₈ * y₃) ^ 2 +
(x₁ * y₇ + x₂ * y₈ + x₃ * y₅ - x₄ * y₆ - x₅ * y₃ + x₆ * y₄ + x₇ * y₁ - x₈ * y₂) ^ 2 +
(x₁ * y₈ - x₂ * y₇ + x₃ * y₆ + x₄ * y₅ - x₅ * y₄ - x₆ * y₃ + x₇ * y₂ + x₈ * y₁) ^ 2
This is Degen's eight squares identity, which describes an intricate relationship between two sums of eight squares each. For any commutative ring `R` and any 16 elements `x₁, x₂, ..., x₈, y₁, y₂, ..., y₈` of `R`, the product of the sum of squares of the `x` variables and the sum of squares of the `y` variables equals the sum of squares of eight specific combinations of the `x` and `y` variables. The specific combinations and their signs correspond to the multiplication of two octonions.
More concisely: For any commutative ring R and any 16 elements x₁, x₂, ..., x₈, y₁, y₂, ..., y₈, the product of (∑xi₂i) and (∑ji₂j) equals ∑(xi∏(δij)⨯yi∏(εij)), where δ and ε are matrices representing multiplication of octonion quaternions.
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