THE CASSINI IDENTITY — a dart

CASSINI'S IDENTITY F(n−1)F(n+1) − F(n)² = ±1 — the invariant that never fades

Multiply the Fibonacci numbers on either side of F(n) and subtract its square: you always get ±1. Exactly. F(n−1)F(n+1) − F(n)² = (−1)ⁿ — Cassini's identity. However large the Fibonacci numbers grow, this combination stays pinned to a unit, flipping sign each step. It is the arithmetic behind the famous “missing square” dissection puzzle (an 8×8 board rearranged into a 5×13 rectangle, gaining one unit of area — because 5·13 − 8² = 1). A conserved quantity riding on top of exponential growth.

THE TECHNIQUE F(n−1)F(n+1) − F(n)² = (−1)ⁿ

The demo computes the Cassini combination across several n — and it is always ±1, whatever the size of the numbers: live demo

HISTORY & CREDIT Giovanni Cassini, 1680

“Products of growing numbers grow.” — this one does not: however big the Fibonaccis, the Cassini combination is always exactly ±1. The famous chessboard-into-rectangle ‘paradox’ is this identity in disguise. cited

1680 · Giovanni Domenico Cassini — the identity (also found by Robert Simson, 1753).
generalization · Catalan's identity extends it to F(n)² − F(n−r)F(n+r).
the puzzle · the 8×8 → 5×13 “missing square” is 5·13 − 8² = 1.

A unit conserved on exponential growth; a paradox that is only an identity. Cassini 1680

RECOMMEND FOR I-13 the conserved unit, on the compiler

On the canonical compiler, for n=5: F(4)·F(6) − F(5)² = 3·8 − 25 = −1 = (−1)⁵:

$ i13 run g_cassini.i13 # F(4)F(6) - F(5)^2 RUN OK · 731 step(s) · peak stack 4 · call depth 6 cassini = -1 -- (-1)^5 ; the combination is pinned to a unit, forever

Recommend: Cassini is a conserved unit riding on exponential growth — i13 computes −1 at n=5, and it would be ±1 at every n. It rhymes with the Kahan keeper (a quantity conserved where you would expect drift) but here the conservation is a theorem of the recurrence, true of every correct Fibonacci — witnessed, not enacted, so not a new keeper. The dart that shows the sequence hides an invariant, and that the “missing square” trick is arithmetic, not magic.