Congruences Modulo s of s-Catalan Numbers
by
Pantelimon Stanica
Auburn University Montgomery

Problems involving binomial coefficients were considered by many mathematicians for over two centuries. R.K. Guy in his Ünsolved Problems in Number Theory" (B31, B33), mentions several problems on divisibility of binomial coefficients. N.J.A. Sloane observed that [3/(5m+3)] binom{5m+3, m} is always an integer and asked for generalizations to [a/r] binom{n, r}, or similar to these. One famous example is that of the Catalan numbers [1/(n+1)] binom{2n, n}. Erdös conjectured that for n > 4, binom{2n, n} is never squarefree. This was proved by Sárközy, for sufficiently large n, and by Granville and Ramaré for any n > 4. Erdös, Graham, Rusza and Straus showed that for any primes p, q there are infinitely many n for which gcd(binom{2n, n}, pq)=1. Hough and the late Simion propose (although they are not the first) the following generalization, which we will call s-Catalan numbers, F(s, n)=[1/((s-1)n+1)] binom{s n, n} and naturally the questions that arise are:

(a) When p is prime, for what values of n is F(p, n) divisible by p?

(b)^* For what values of n is F(4, n) divisible by 4?

(c)^* What can you say when s takes on the other composite values?

There are no known answers for (b), (c). We plan on changing that.

Date received: March 28, 2000