Problem Statement
For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\leq i<k$ such that $n-i$ is $k$-smooth, that is, divisible only by primes $\leq k$.
Are there infinitely many binomial coefficients with deficiency $1$? Are there only finitely many with deficiency $>1$?
Are there infinitely many binomial coefficients with deficiency $1$? Are there only finitely many with deficiency $>1$?
Categories:
Number Theory Binomial Coefficients
Progress
A problem of Erdős, Lacampagne, and Selfridge [ELS88], that was also asked in the 1986 problem session of West Coast Number Theory (as reported here).In [ELS93] they prove that if the deficiency exists and is $\geq 1$ then $n\ll 2^k\sqrt{k}$.
The following examples are either from [ELS88] or here. The following have deficiency $1$ (there are $58$ examples with $n\leq 10^5$):\[\binom{7}{3},\binom{13}{4},\binom{14}{4},\binom{23}{5},\binom{62}{6},\binom{94}{10},\binom{95}{10}.\]The examples which follow are the only known examples with deficiency $>1$. The following have deficiency $2$:\[\binom{44}{8},\binom{74}{10},\binom{174}{12},\binom{239}{14},\binom{5179}{27},\binom{8413}{28},\binom{8414}{28},\binom{96622}{42}.\]The following have deficiency $3$:\[\binom{46}{10},\binom{47}{10},\binom{241}{16},\binom{2105}{25},\binom{1119}{27},\binom{6459}{33}.\]The following has deficiency $4$:\[\binom{47}{11}.\]The following has deficiency $9$:\[\binom{284}{28}.\]See also [384] and [1094].
Barreto in the comments has given a positive answer to the second question, conditional on two (strong) conjectures.
Source: erdosproblems.com/1093 | Last verified: January 19, 2026