Products of factorials in binary recurrence sequences
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.
non-degenerate binary recurrence sequence
linear forms in logarithms of algebraic numbers
scientific article; zbMATH DE number 1455475