Let be a prime, and let be the Fermat quotient of to base . The following curious congruence was conjectured by L. Skula and proved by A. Granville In this note we establish the above congruence by entirely elementary number theory arguments.
By employing one of the cubic transformations (due to W. N. Bailey (1928)) for the -series, we examine a class of -series. Several closed formulae are established by means of differentiation, integration and contiguous relations. As applications, some remarkable binomial sums are explicitly evaluated, including one proposed recently as an open problem.
A positive integer is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number .
In this paper, we apply character sum estimates to study products of factorials modulo .