Lucas Squares and Twice Squares
Abstract
Dyadic Lucas moduli classify every square and twice-square Lucas value.
Theorem 1.1 (Complete classifications, including index zero).
Lean statement: D5/S3/Arith/Primes/LucasSquareClassification.lucas_square_classifications
Proof. Machine-checked in Lean as D5/S3/Arith/Primes/LucasSquareClassification.lucas_square_classifications (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every natural index n, the integer Lucas number L_n is a square exactly when n is one or three. It equals twice an integer square exactly when n is zero or six. The exceptional values are L_0 = 2, L_1 = 1, L_3 = 4 and L_6 = 18.
The proof follows Cohn’s classical classifications in Square Fibonacci Numbers, Etc., Theorems 1 and 2. Finite reductions modulo four exclude all even indices from the square case; reductions modulo eight exclude all odd indices from the twice-square case. The remaining indices use a modulus L_(2^(r+1)), which is three modulo four.
Trace and norm in the golden integer ring give an antiperiod of the golden unit modulo that Lucas number. An odd multiple of the antiperiod makes the required residues minus one, minus four, or minus thirty-six. In the last case, the modulus is also coprime to three. Their Jacobi symbols are minus one. The residue obtained through the negative index minus six is derived by multiplying by the conjugate of the sixth power.
These are the classical one-index Lucas classifications used by the Fibonacci square-class argument. General product exclusions for distinct Fibonacci or Lucas indices require their separate proofs.