An integer  is -balanced for  a prime if, among all nonzero binomial coefficients  for , ...,  (mod ), there are equal numbers of quadratic residues and nonresidues (mod ). Let  be the set of integers , , that are -balanced. Among all the primes , only those with , 3, and 11 have .

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

2
3
5	{3}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline22.gif" style="height:15px; width:17px" />
7	{3}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline23.gif" style="height:15px; width:17px" />
11
13	{7,11}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline25.gif" style="height:15px; width:39px" />
17	{3,15}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline26.gif" style="height:15px; width:39px" />

REFERENCES:

Garfield, R. and Wilf, H. S. "The Distribution of the Binomial Coefficients Modulo ." 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.