Work of T . N . Shorey 3 2 Applications of Linear Form Estimates to Values of Polynomials , Recurrence Sequen

We state a number of important results which we owe to Tarlok Shorey. 1 Shorey’s Contributions to Linear Form Estimates and Some Applications One of the first results of Shorey concerns a sharpening of a theorem of Sylvester. Sylvester proved in 1892 that a product of k consecutive positive integers greater than k is divisible by a prime exceeding k. By combining a result of Jutila which depends on estimates for exponential sums and an estimate on linear forms in logarithms, Shorey [45] proved in 1974 that it suffices to take constant times k(log logk)/logk consecutive integers in place of k consecutive integers in the above result of Sylvester. This improved on results of Erdős, Tijdeman, and Ramachandra and Shorey and is still the best known. The used estimate for the linear form itself is an important contribution of Shorey to the theory on estimating linear forms in logarithms of algebraic numbers which had been developed by Baker in the preceding decade. Since estimates on linear forms play an important role in Shorey’s work, we state his result. If a and b are coprime integers then the size of the rational number a/b is defined as |b|+ |a/b|. All the constants C1, C2, . . . appearing in this article are effectively computable. This means that they can be determined explicitly in terms of the various parameters under consideration. Let n > 1 be an integer. Let α1 = m m′ , α2 = p2 p′2 , . . . , αn = pn pn