The th central binomial coefficient is defined as
, where is a Binomial
Coefficient and
is the Floor Function. The first few values are 1, 2, 3, 6, 10, 20, 35, 70, 126, 252,
... (Sloane's A001405). The central binomial coefficients have Generating Function

The central binomial coefficients are Squarefree only for , 2, 3, 4, 5, 7, 8, 11, 17, 19, 23, 71, ... (Sloane's A046098), with no others less than 7320.

The above coefficients are a superset of the alternative ``central'' binomial coefficients

which have Generating Function

The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (Sloane's A000984).

Erdös and Graham (1980, p. 71) conjectured that the central binomial coefficient is *never*
Squarefree for , and this is sometimes known as the Erdös Squarefree Conjecture. Sárközy's Theorem (Sárközy 1985) provides a partial solution which states
that the Binomial Coefficient is never Squarefree for all sufficiently large (Vardi
1991). Granville and Ramare (1996) proved that the *only* Squarefree values are and 4. Sander (1992)
subsequently showed that
are also never Squarefree for sufficiently large as long as is
not ``too big.''

**References**

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.

Sárközy, A. ``On Divisors of Binomial Coefficients. I.'' *J. Number Th.* **20**, 70-80, 1985.

Sloane, N. J. A. Sequences
A046098,
A000984/M1645, and
A001405/M0769,
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Vardi, I. ``Application to Binomial Coefficients,'' ``Binomial Coefficients,'' ``A Class of Solutions,''
``Computing Binomial Coefficients,'' and ``Binomials Modulo and Integer.''
§2.2, 4.1, 4.2, 4.3, and 4.4 in *Computational Recreations in Mathematica.*
Redwood City, CA: Addison-Wesley, pp. 25-28 and 63-71, 1991.

© 1996-9

1999-05-26