Given binomial coefficient , write

for , where contains only those prime factors . Then the number of for which (i.e., for which all the factors of are is called the deficiency of (Erdős et al. 1993, Guy 1994). The following table gives the good binomial coefficients (i.e., those with ) having deficiency (Erdős et al. 1993), and Erdős et al. (1993) conjecture that there are no other with .

	good binomial coefficients
1	, , , , , , , ...
2	, , , , , ,
	,
3	, , , , ,
4
9

Erdős, P.; Lacampagne, C. B.; and Selfridge, J. L. "Estimates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215-224, 1993.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 84-85, 1994.

CITE THIS AS:

Weisstein, Eric W. "Deficiency." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Deficiency.html