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Nonsquares in Higher Odd Prime-Power Layers

Abstract

All higher odd prime-power Fibonacci quotients are positive nonsquares beyond an explicit prime threshold.

Theorem 1.1 (An analytic obstruction for recurrence polynomials and Fibonacci quotients).

Lean statement: D5/S3/Arith/Primes/FibonacciRecurrencePolynomialNonsquare.fibonacci_recurrence_polynomial_nonsquare

Proof. Machine-checked in Lean as D5/S3/Arith/Primes/FibonacciRecurrencePolynomialNonsquare.fibonacci_recurrence_polynomial_nonsquare (✓ std3). ∎

Source. Repository-derived.

Commentary.

For r at least 119, the recurrence polynomial U_(2r+1) takes a nonsquare value at every integer x greater than two satisfying x^2 - 4 > 128 times 6^r. Pairing its complex roots yields factors X^2 + lambda_i with 0 at most lambda_i less than four. Reversing and contracting the recurrence polynomial produces an integral polynomial whose linear coefficient is odd. The half-binomial expansion then has odd dyadic coefficient numerators; its analytic square-root branch obeys a Cauchy coefficient bound. The dyadic analytic integer obstruction excludes an integer square root. For every odd n at least 2r+1, the Fibonacci quotient F_((2r+1)n)/F_n is positive and nonsquare. In particular, for every prime q at least 239 and every k at least one, F_(q^(k+1))/F_(q^k) is positive and nonsquare.

References