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**Products of factorials in binary recurrence sequences.**
*(English)*
Zbl 0978.11010

The author shows that every non-degenerate binary recurrence sequence contains only finitely many terms which can be written as product of factorials. Further these terms can be effectively computed. In particular for the Lucas sequence \((L_n)_{n\geq 0}\), \(L_0=2\), \(L_1=1\) and \(L_{n+2}= L_{n+1}+ L_n\), the only terms which are products of factorials are \(L_0= 2!\) and \(L_3= (2!)^2\). The corresponding result for the Fibonacci sequence \((F_n)_{n\geq 0}\), \(F_0=0\), \(F_1=1\) and \(F_{n+2}= F_{n+1}+ F_n\), is \(F_3= 2!\), \(F_6= (2!)^3\) and \(F_{12}= (2!)^2 (3!)^2= 3!4!\). The proofs use estimations of linear forms in logarithms of algebraic numbers.

Reviewer: N.Saradha (Mumbai)

### MSC:

11D61 | Exponential Diophantine equations |

11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |

11J86 | Linear forms in logarithms; Baker’s method |

### Keywords:

exponential Diophantine equations; non-degenerate binary recurrence sequence; product of factorials; Lucas sequence; Fibonacci sequence; linear forms in logarithms of algebraic numbers### References:

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