Indag. Mathem., N.S., 1 (4), 465-472 December 17, 1990 Some Ramanujan-Nagell equations with many solutions by P. Moree’ and C.L. Stewart2 ’ Department of Mathematics and Computer Science, University of Leiden, P.O. Box 9512, 2300 RA Leiden, the Netherlands ’ Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI Communicated by Prof. R. Tijdeman at the meeting of May 28, 1990 1. INTRODUCTION Let F(x, y) be a binary form with integer coefficients of degree nl3 and let S= {Pi, ..*, pz} be a set of prime numbers. In 1984 Ever&e [5] proved that if the binary form zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F is divisible by at least three pairwise linearly independent linear forms in some algebraic number field then the number of solutions of (1) F(x, y) = pf’ . . . p," , in coprime integers x and y and integers zi, . . . ,z, is at most (2) 2 x +s + 3) Equation (1) is known as a Thue-Mahler equation. Estimates for the number of solutions of (1) had been given earlier by Mahler [12] and Lewis and Mahler [ 111. Recently Bombieri [l] proved that if F is of degree at least 6 and is without multiple factors then the number of solutions of (1) in coprime integers x and y and integers zi, . . . , z, is at most (3) (4(s + 1))2 (4n)*qS+ l). If we fix y as 1 in (1) we obtain a Ramanujan-Nagell equation. In [4] Erdos, Stewart and Tijdeman proved that the exponential dependence on s in estimates (2) and (3) is not far from the truth by giving examples of Ramanujan-Nagell equations with many solutions. Let E be a positive number, let 2=pl,p2, . . . be the sequence of prime numbers and let n be an integer with n z 2. They proved 465