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Balanced Binomial Coefficient

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An integer n is p-balanced for p a prime if, among all nonzero binomial coefficients (n; k) for k=0, ..., n (mod p), there are equal numbers of quadratic residues and nonresidues (mod p). Let T_p be the set of integers n, 0<=n<=p-1, that are p-balanced. Among all the primes <1000000, only those with p=2, 3, and 11 have T_p=emptyset.

The following table gives the p-balanced integers for small primes p (OEIS A093755).

pT_p
2emptyset
3emptyset
5{3}
7{3}
11emptyset
13{7,11}
17{3,15}

See also

Binomial Coefficient

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References

Garfield, R. and Wilf, H. S. "The Distribution of the Binomial Coefficients Modulo p." J. Number Th. 41, 1-5, 1992.Sloane, N. J. A. Sequence A093755 in "The On-Line Encyclopedia of Integer Sequences."Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdős. Papers from the Conference in Honor of Erdős' 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bollobás and A. Thomason). Cambridge, England: Cambridge University Press, pp. 557-562, 1997.

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Balanced Binomial Coefficient

Cite this as:

Weisstein, Eric W. "Balanced Binomial Coefficient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BalancedBinomialCoefficient.html

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