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Erdös Squarefree Conjecture

The Central Binomial Coefficient ${2n\choose n}$ is never Squarefree for $n>4$. This was proved true for all sufficiently large $n$ by Sárközy's Theorem. Goetgheluck (1988) proved the Conjecture true for $4<n\leq 2^{42205184}$ and Vardi (1991) for $4<n<2^{774840978}$. The conjecture was proved true in its entirely by Granville and Ramare (1996).

See also Central Binomial Coefficient


Erdös, 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.

Sárközy, 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.

© 1996-9 Eric W. Weisstein