##
**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 \(| t| ^{\prime\prime}\).

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 \(| t| ^{\prime\prime}\).

Reviewer: Wolfgang Schwarz (Frankfurt / Main)

### 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) |

### Keywords:

exponential Diophantine equations; lower bounds for linear forms in logarithms; Thue equation; elliptic curves; perfect powers in the Fibonacci sequence; Lucas sequence; ideas from Wiles’ proof of Fermat’s conjecture; upper bounds for solutions of Thue equations; regulator of algebraic number fields### Citations:

Zbl 0843.11036
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XMLCite

\textit{Y. Bugeaud} et al., Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)

### Online Encyclopedia of Integer Sequences:

Fibonacci numbers which are perfect powers.Fibonacci numbers that are the product of other Fibonacci numbers.