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Sárkőzy's Theorem

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. Sárkőzy (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.Sárkőzy, 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.

Referenced on Wolfram|Alpha

Sárkőzy's Theorem

Cite this as:

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

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