On two conjectures of Erdös
by
Florian Luca
Mathematics Department, Bielefeld University, 33501 Bielefeld, Germany

Conjecture 1 (Erdös and Stewart): For any k >= 1, let p_k be the kth prime number. Erdös and Stewart asked for all solutions of the equation n! +1=p_k^ap_k+1^b when p_k-1 <= n < p_k. In the first part of the talk, I'll conform their conjecture which asserts that the largest solution of the above equation is 5!+1=11². The proof uses a linear form in two logarithms due to Bugeaud and Laurent as well as a result of Erdös and Oblath on factorials of a certain type.

Conjecture 2 (Erdös and Graham): What are the perfect powers in the Pascal triangle ? That is, which binomial coefficients are perfect powers ? Erdös and Oblath treated various cases of the above problem leaving a few instances unsolved. In the second part of my talk, I'll completely treat the remaining cases. The proof uses some recent results of Bennet, Darmon, Merel and de Weger on the Siegel equation |axⁿ -byⁿ|=1.

Date received: March 9, 1999