Sárközy's Theorem

A partial solution to the Erdös Squarefree Conjecture which states that the Binomial Coefficient ${2n\choose n}$ is never Squarefree for all sufficiently large $n\geq n_0$. Sárközy (1985) showed that if $s(n)$ is the square part of the Binomial Coefficient ${2n\choose n}$, then

$$\ln s(n)\sim (\sqrt{2}-2)\zeta({\textstyle{1\over 2}})\sqrt{n},$$

where $\zeta(z)$ is the Riemann Zeta Function. An upper bound on $n_0$ of $2^{8,000}$ has been obtained.

See also Binomial Coefficient, Erdös Squarefree Conjecture

References

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, 1980.

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

Sárközy, A. ``On the 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.