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Complete Prime-Power Fibonacci Layers

Abstract

Every nontrivial prime-power Fibonacci quotient away from five has exact fresh entry ranks.

Theorem 1.1 (Fresh ranks and original depths for every prime factor).

Lean statement: D5/S3/Arith/Primes/FibonacciPrimePowerLayer.fibonacci_prime_power_layer

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

Source. Repository-derived.

Commentary.

Let q be a prime different from five and let s be positive, excluding q = 2, s = 1. The natural quotient C = F_(q^s)/F_(q^(s-1)) is greater than one, is coprime to the denominator Fibonacci value, and is not a square. Every prime p dividing C has first Fibonacci entry rank q^s, differs from q, and has the same valuation in C as in the Fibonacci number at its first entry rank. At least one such prime has odd original depth.

For the dyadic layers, every factor is odd because the entry rank of two is three. A factor with an earlier rank would divide both adjacent Fibonacci values. Since it is prime to their indices, the original-rank valuation law would give equal valuations in those values, contradicting its positive depth in the quotient. The cubic-block rank theorem handles the higher ternary layers, while the local entry-rank and valuation calculation handles primes at least seven.

Exact rank excludes old factors and yields coprimality. The dyadic and cubic congruences and the odd-prime quotient estimates exclude squares; factorization then supplies an odd-depth prime. The omitted dyadic quotient equals one. Five is excluded because F_25/F_5 still shares a factor five with F_5.

References