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Sárközy's Theorem

Sárközy's theorem is a partial solution to the Erdős squarefree conjecture which states that the binomial coefficient (2n; n) is never squarefree for all sufficiently large n>=n_0. Sárközy (1985) showed that if s(n) is the square part of the binomial coefficient (2n; n), then

lns(n)∼(sqrt(2)-2)zeta(1/2)sqrt(n),

where zeta(z) is the Riemann zeta function. An upper bound on n_0 of 2^(8000) has been obtained.

See also

Binomial Coefficient, Erdős Squarefree Conjecture

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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, 1980.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. https://doi.org/10.1016/0022-314X(85)90017-4.Vardi, I. "Applications to Binomial Coefficients." Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 25-28, 1991.

Cite this as:

Weisstein, Eric W. "Sárközy's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SarkoezysTheorem.html

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