## Classical and modular approaches to exponential Diophantine equations. I: Fibonacci and Lucas perfect powers.(English)Zbl 1113.11021

The authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equation, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem.
The authors are going to
– present theoretical improvements to various aspects of the classical approach (through linear forms in logarithms),
– show how local information obtained through the modular approach (associate to a maybe – existing solution of a Diophantine equation an elliptic curve) can be used to reduce the size of the bounds [for exponents and variables], and to show how this local information can be pieced together to prove that there are no missing solutions,
– solve various outstanding Diophantine equations.
Using the described strategy, the authors show (after a short historical survey of known results) two spectacular results:
The only perfect powers in the Fibonacci sequence $$F_n$$ [$$F_0=0, F_1=1, F_{n+2} = F_{n+1} + F_n$$] are $$F_0=0,\; F_1=1,\; F_2=1,\; F_6=8$$ and $$F_{12}=144$$,
and, the only perfect powers in the Lucas sequence $$L_n$$ [$$L_0=2,\; L_1=1,\;$$ and $$L_{n+2} = L_{n+1} + L_n$$] are $$L_1=1$$ and $$L_3=4$$.
The authors describe in a very lucid way the eight main steps for deducing these results. Relying in part on a paper of M. Laurent, M. Mignotte and Y. Nesterenko [J. Number Theory 55, No. 2, 285–321 (1995; Zbl 0843.11036)], the authors prove an important improvement of lower bounds for linear forms in three logarithms, $\Lambda = b_2 \log \alpha_2 - b_1 \log \alpha_1 - b_3\log \alpha_3,$ where $$b_j$$ are positive rational integers with gcd$$(b_1,b_2,b_3)=1$$, and the $$\alpha_j$$ are non-zero multiplicatively independent algebraic numbers. Given $$\rho \geq e$$, and assume that several inequalities concerning the integers $$K, L$$, $$R, R_i$$, $$S, S_i, T, T_i$$, all $$\geq 3$$, are fulfilled. Then either $| \Lambda| \cdot \max\left\{ \frac{LRe^{LR| \Lambda| /(2b_1)}}{2b_1}, \frac{LSe^{LS| \Lambda| /(2b_2)}}{2b_2}, \frac{LTe^{LT| \Lambda| /(2b_3)}}{2b_3} \right\} > \rho^{-KL}$ is true or one of the conditions (C1), (C2), (C3) holds. Here (C1) is $\exists r,s\in\mathbb{Z}, \; rb_2 = sb_1 \text{ with } 0<r\leq R_i, 0<s\leq S_i \text{ for some } i=1,2,$ (C2) is similar, and (C3) means $\exists r^\prime, s^\prime, t^\prime, t^{\prime\prime} \in \mathbb{Z} \text{ such that } s^\prime t^\prime b_1 + r^\prime t^{\prime\prime} b_2 + r^\prime s^\prime b_3 = 0,$ with bounds $0<| r^\prime| < \min\left\{ R_1 +1, \left(\frac{(R_1+1)(S_1+1)}{T_1+1}\right)^{\frac12}\right\}$ and similar bounds for $$| s^\prime| ,\; | t| ^\prime$$, and $$| t| ^{\prime\prime}$$.

### MSC:

 11D61 Exponential Diophantine equations 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11J86 Linear forms in logarithms; Baker’s method 11G99 Arithmetic algebraic geometry (Diophantine geometry)

Zbl 0843.11036

PARI/GP; Magma
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