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).
Erdős Squarefree Conjecture
See also
Central Binomial CoefficientExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Erdős Squarefree ConjectureCite this as:
Weisstein, Eric W. "Erdős Squarefree Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html