Erdős Squarefree Conjecture

The central binomial coefficient  is never squarefree for . This was proved true for all sufficiently large  by Sárkőzy's theorem. Goetgheluck (1988) proved the conjecture true for  and Vardi (1991) for . The conjecture was proved true in its entirety by Granville and Ramare (1996).

REFERENCES:

Erdős, P. and Graham, R. L. Old and New Problems and Results in Combinatorial Number Theory. Geneva, Switzerland: L'Enseignement Mathématique Université de Genève, Vol. 28, p. 71, 1980.

Goetgheluck, P. "Prime Divisors of Binomial Coefficients." Math. Comput. 51, 325-329, 1988.

Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.

Sander, J. W. "On Prime Divisors of Binomial Coefficients." Bull. London Math. Soc. 24, 140-142, 1992.

Sander, J. W. "A Story of Binomial Coefficients and Primes." Amer. Math. Monthly 102, 802-807, 1995.

Sárkőzy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985.

Vardi, I. "Applications to Binomial Coefficients." Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 25-28, 1991.