Odd multiplicity and log-concavity in Fibonacci quotients
Year: 2026
Type: article
Abstract: This paper studies two questions about Fibonacci quotients: the occurrence of prime divisors of odd multiplicity at semiprime indices, and the log-concavity of the coefficients of q-Fibonomial polynomials. It combines exact arithmetic identities, congruence methods, effective Diophantine bounds, polynomial positivity certificates, and quantitative local limit estimates. The principal coefficient result is a uniform theorem for all parameter pairs with both parameters at least 1024. Together with effective stabilization in every fixed column from four onward, this reduces the complete coefficient classification to an effectively bounded finite family. For Fibonacci numbers indexed by the product of two distinct primes, the arithmetic analysis identifies the exact part contributed by primes appearing for the first time at that index. The usual cyclotomic quotient can contain one older prime factor: the larger index prime, when it divides the Fibonacci number at the smaller prime index. Removing this factor gives an exact integer core. The existence of a first-appearance divisor of odd multiplicity is equivalent to this core being nonsquare. Six compatible periodic quadratic-residue tests certify nonsquareness on an explicit family with relative prime-pair density 41/48, where both primes are bounded by the same growing parameter. This is the density of the specified certified family and a lower bound for the proportion of all successful pairs in that ordering. A separate adaptive congruence proves that every prime multiplier at least thirteen has a successful collision partner. It also resolves every collision for which the Fibonacci number at the smaller prime index is a prime times a square. An effective hyperelliptic height theorem gives an explicit bound on the larger prime in terms of the smaller prime. Every partner above this bound has the required odd-multiplicity divisor. For each fixed smaller prime, possible failures therefore lie in a finite interval. A Lucas-doubling criterion supplies a complementary finite orbit of Jacobi-symbol tests. Exact witnesses are recorded for all 4,005 pairs of distinct primes from thirteen through five hundred, including all fourteen collision pairs in that range. The smaller prime remains unbounded in the unrestricted problem. The case of indices thirteen times a prime admits a more detailed geometric reduction. Compatible local conditions select a single elliptic cover over the quadratic field generated by the square root of thirteen. The paper proves that this curve has trivial torsion and exhibits two independent points through an elementary argument using real components and finite-field reduction. Its rank is consequently at least two, placing this cover outside the usual rank-below-field-degree condition for elliptic Chabauty. Excluding the required Lucas-square coordinates remains an open arithmetic problem. For q-Fibonomial polynomials, the paper proves the complete coefficient log-concavity classification whenever the smaller parameter is at most fourteen. Up to exchanging the parameters, the exceptions in this range are column three when the larger parameter is congruent to zero or one modulo three, and the pair with both parameters equal to four. The proofs use exact generating-function identities, bounds for periodic terms, and rational Bernstein certificates on unbounded parameter domains, together with exact calculations on the finite intervals below their thresholds. A second theorem proves log-concavity at every index for every Fibonacci denominator kernel with at least four factors. The equality index is one for kernels with four or five factors, and the equality indices are one and seven for all larger kernels. The infinite kernel, including both weight-one factors, has the same latter equality set. This uniform result combines exact polynomial certificates with a centered Gamma comparison on a tilted contour. The polynomial analysis then establishes eventual log-concavity in every fixed column from four onward with an effectively computable threshold. A global strong log-concavity estimate for the continuous convolution of the actual numerator boxes, an explicit transfer through the denominator deconvolution, and Fourier tail bounds yield the uniform cutoff 1024. Above this cutoff, every interior coefficient determinant is strictly positive except at distances one and seven from the two endpoints, where equality holds exactly. Further results give explicit central and outer concavity intervals and certify the first and last one million coefficient positions uniformly from column eleven onward. Four complete parameter pairs, fifteen and fifteen, sixteen and fifteen, seventeen and fifteen, and sixteen and sixteen, are verified by 8,545,172 exact lower-half determinant calculations, with reciprocity covering the upper halves. Structural examples also identify limitations of log-concave factorization, higher-order total positivity, and estimates based only on inherited log-concavity under numerator activation. The complete coefficient conjecture remains open only in the effectively bounded parameter ranges within columns fifteen through 1023 below their fixed-column thresholds, excluding the cases and coefficient intervals already certified. The unrestricted semiprime conjecture also remains open: effective bounds for each smaller prime and the finite Lucas witness computations do not establish a theorem for every prime pair. The accompanying project contains the LaTeX manuscript, compiled paper, exact certificate data, verification programs, independent reconstruction routines, and a 46-stage reproduction driver. The computations use integer and rational arithmetic, together with outward-rounded Arb interval bounds where analytic inequalities require certified enclosures. Finite computations, inequalities on unbounded domains, and open conjectures are distinguished throughout.
Author: Fathi, Kevin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213752903
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