## 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.
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