| We have examined a collection of Diophantine equations relating the classical Fibonacci sequence {lcub}Fm{rcub} and sums of consecutive or near consecutive factorials. This problem originated in the area of combinatorial number theory and its solution involves both combinatorial and algebraic number theory. Additional groundwork for the solution lies in the theory of Fibonacci numbers and of linear forms in logarithms. A series of related problems has recently been solved, and the present work serves to extend these solutions.; A collection of techniques from algebraic and computational number theory was used, in particular a recent theorem which gives bounds on the powers of prime divisors of differences of algebraic numbers taken to rational integer powers. Additional techniques used include primitive divisors, periodicity of the Fibonacci sequence modulo various primes, direct computation, and a very recent result concerning bounds for binary recurrence sequences using lower bounds for linear forms in logarithms of algebraic numbers. In the equation Fm = n! + (n + 1)! + … + (n + k)!, we have shown that there are at most finitely many solutions for a given value of k, as n is bounded above by a function of k. In addition, we have shown by direct calculation that the only solution with 2 ≤ k ≤ 10 is F9 = 34 = 0! + 1! + 2! + 3! + 4!. We have established upper bounds on the size of solutions to the equation Fm = ±n 1! ± n2! + n 3!: if n1 ≥ 4, then m < e53. Additionally, we have determined almost all of the exact solutions to equations involving Fibonacci numbers and sums of three factorials with two of them consecutive. Current methods have been used to their fullest possible extent, and it appears that new methods will be needed to extend the solutions further. Two interesting conjectures are also posed which relate to a more general problem of consecutive factorials and Fibonacci numbers. |
