TOPICS
Search

Deficiency


Given binomial coefficient (N; k), write

 N-k+i=a_ib_i,

for 1<=i<=k, where b_i contains only those prime factors >k. Then the number of i for which b_i=1 (i.e., for which all the factors of N-k+i are <=k is called the deficiency of (N; k) (Erdős et al. 1993, Guy 1994). The following table gives the good binomial coefficients (i.e., those with lpf(N; k)>k) having deficiency d>=1 (Erdős et al. 1993), and Erdős et al. (1993) conjecture that there are no other with d>1.

dgood binomial coefficients
1(3; 2), (7; 3), (13; 4), (14; 4), (23; 5), (62; 6), (89; 8), ...
2(7; 4), (44; 8), (74; 10), (174; 12), (239; 14), (5179; 27),
(8413; 28), (96622; 42)
3(46; 10), (47; 10), (241; 16), (2105; 25), (1119; 27), (6459; 33)
4(47; 11)
9(284; 28)

See also

Abundance, Good Binomial Coefficient

Explore with Wolfram|Alpha

References

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.

Referenced on Wolfram|Alpha

Deficiency

Cite this as:

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

Subject classifications