Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 746.11021
Autor: Brindza, B.; Erdös, Paul
Title: On some diophantine problems involving powers and factorials. (In English)
Source: J. Aust. Math. Soc., Ser. A 51, No.1, 1-7 (1991).
Review: An old conjecture states that any solution in positive integers n and x of the equation 1+n! = x2 is given by n = 4,5,7. In this paper it is shown that for every r in N there is an n0 = n0(r) such that none of the integers sumri = 1ni! (n0 < n1 < ... < nr) is powerful, i.e. not all exponents of the primes occurring in the prime factorization of such an integer are larger than 1. Unfortunately, the n0(r) can not be given explicitly. The authors also show that there is an effectively computable upper bound for the solution in positive integers a, k, p (p > 2 and prime) of the equation (p-1)!+ap-1 = pk. The proof depends on deep results on linear forms in logarithms. Finally the same method is applied to obtain an effectively computable upper bound for k in a solution in positive integers x, k, p (p > 1) of the Ramanujan-Nagell equation x2+D = pk (D\ne0, D in Z) which is close to being best possible in D.
Reviewer: R.J.Stroeker (Rotterdam)
Classif.: * 11D61 Exponential diophantine equations
Keywords: factorial diophantine equation; Baker's method; upper bound; Ramanujan- Nagell equation; power values of sum of factorials
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