The Erdős squarefree conjecture states that 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). The conjecture is problem 175
in the Erdős problems collection (Bloom 2026).
Erdős Squarefree Conjecture
See also
Central Binomial Coefficient, Erdős ProblemsExplore with Wolfram|Alpha
References
Bloom, T. F. "Erdős Problem 175." Erdős Problems. Oct. 1, 2026. https://www.erdosproblems.com/175.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 Resource. https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html