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Date: 5-10-2020
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Date: 13-8-2020
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Date: 27-10-2019
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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 | |
7 | |
11 | |
13 | |
17 |
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.
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