Balanced Binomial Coefficient
المؤلف:
Sloane, N. J. A.
المصدر:
Sequence A093755 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
19-10-2020
957
Balanced Binomial Coefficient
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 |
![<span style=]() {3}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline22.gif" style="height:15px; width:17px" /> |
| 7 |
![<span style=]() {3}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline23.gif" style="height:15px; width:17px" /> |
| 11 |
 |
| 13 |
![<span style=]() {7,11}" src="https://mathworld.wolfram.com/images/equations/BalancedBinomialCoefficient/Inline25.gif" style="height:15px; width:39px" /> |
| 17 |
![<span style=]() {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.
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