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Product of powers of distinct primes as sums of Fibonacci numbers

Herbert Batte, Florian Luca, Volker Ziegler

TL;DR

This work determines all prime pairs $(p,q)$ with $q\le 1000$ for which the Fibonacci-sum Diophantine equation $F_n+F_m=p^xq^y$ admits at least two distinct $(x,y)$-solutions with $n\ge m$, excluding degeneracies. The authors combine Zeckendorf decomposition, Baker-type lower bounds for linear forms in logarithms, and LLL lattice reduction, together with continued-fraction refinements and targeted computational checks, to bound $n$ and exhaust all cases. They prove that only six pairs $(p,q)\in\{(3,2),(5,2),(7,2),(7,3),(17,2),(19,2)\}$ yield at least two representations, and they list the corresponding Fibonacci-sum decompositions explicitly. The results illustrate how analytic number theory tools plus computational verification can completely resolve a constrained exponential Diophantine problem involving sums of Fibonacci numbers, with potential implications for related additive-multiplicative Fibonacci identities.

Abstract

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes $(q,p)$ with $q\le \min\{1000,p\}$ such that the above equation has at least two solutions $(x,y)$ (and corresponding $m,n$) in positive integers.

Product of powers of distinct primes as sums of Fibonacci numbers

TL;DR

This work determines all prime pairs with for which the Fibonacci-sum Diophantine equation admits at least two distinct -solutions with , excluding degeneracies. The authors combine Zeckendorf decomposition, Baker-type lower bounds for linear forms in logarithms, and LLL lattice reduction, together with continued-fraction refinements and targeted computational checks, to bound and exhaust all cases. They prove that only six pairs yield at least two representations, and they list the corresponding Fibonacci-sum decompositions explicitly. The results illustrate how analytic number theory tools plus computational verification can completely resolve a constrained exponential Diophantine problem involving sums of Fibonacci numbers, with potential implications for related additive-multiplicative Fibonacci identities.

Abstract

Let be the -th Fibonacci number. In this paper, we study the Diophantine equation in nonnegative integers , and , where and are fixed distinct prime numbers. We determine all pairs of primes with such that the above equation has at least two solutions (and corresponding ) in positive integers.
Paper Structure (22 sections, 14 theorems, 311 equations)

This paper contains 22 sections, 14 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Methods
  5. Preliminaries on Fibonacci and Lucas numbers
  6. Generalities on the equation $F_n+F_m=p^xq^y$
  7. Linear forms in logarithms
  8. Reduction methods
  9. Continued fractions
  10. Reduced Bases for Lattices and LLL--reduction methods
  11. Proof of Theorem \ref{['th:main']}
  12. Reduction to Zeckendorf representations
  13. An absolute upper bound on $n$ knowing $p$ and $q$
  14. The case for small primes $2\le q<p\le 1000$
  15. The case $q\le 1000$ and $p>1000$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $p>q$ be fixed distinct primes with $q\le 1000$. Then the Diophantine equation has at least two distinct solutions $(x,y)$ only when $(p,q)\in \mathcal{S}$, where In this case, the sums of two Fibonacci numbers that factor as $p^xq^y$ for positive integers $x$ and $y$, with $(p,q)\in \mathcal{S}$, are as follows:

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 2.1: Carmichael
  • Lemma 2.1: McDaniel, MD
  • Theorem 2.2: Bugeaud, Mignotte, Siksek
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.1
  • Theorem 2.3: Matveev, Theorem 9.4 in Buge
  • ...and 10 more