Cassini's identity

Cassini's identity states that consecutive Fibonacci numbers on either side of a term satisfy Fₙ₋₁Fₙ₊₁ − Fₙ² = (−1)ⁿ for n ≥ 1, with F₀ = 0 and F₁ = 1.

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Fibonacci sequence: The sequence 0, 1, 1, 2, 3, 5, …, in which each term after the first two is the sum of its predecessors. Its recurrence supplies all terms in Cassini’s identity.
Fibonacci number: A term Fₙ of the Fibonacci sequence, defined by F₀ = 0, F₁ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂. Cassini’s identity relates three Fibonacci numbers with consecutive indices.
Giovanni Domenico Cassini: An Italian-born astronomer and mathematician who directed the Paris Observatory and studied Saturn’s moons and rings. The identity is named after Cassini, who stated it in the seventeenth century.
Fibonacci divisibility: The study of divisibility relations among Fibonacci numbers, including when one Fibonacci number divides another. Cassini’s identity supplies congruences useful in divisibility arguments.
Catalan's identity: An identity expressing Fₙ₋ₖFₙ₊ₖ − Fₙ² in terms of Fibonacci numbers at index k. Setting k = 1 recovers Cassini’s identity.
Fibonacci recurrence: The rule Fₙ = Fₙ₋₁ + Fₙ₋₂ defining each Fibonacci number from the two preceding numbers. Substituting this rule turns the identity’s left side into a simpler expression.
Integer sequence: An ordered list of integers indexed by whole numbers or another discrete set. Cassini’s identity is a relation among terms of a particular integer sequence.
Fibonacci: Leonardo of Pisa, the medieval mathematician whose Liber Abaci helped introduce Hindu-Arabic numerals to Europe. The sequence central to Cassini’s identity bears Fibonacci’s name.
Fibonacci numbers modulo m: The Fibonacci sequence considered by its remainders after division by a fixed integer m. Reducing Cassini’s identity modulo m yields constraints on these remainders.
d'Ocagne's identity: The relation FₘFₙ₊₁ − Fₘ₊₁Fₙ = (−1)ᵐFₙ₋ₘ for Fibonacci numbers. It generalizes the neighboring-index determinant pattern behind Cassini’s formula.