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.

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This page was last edited 27 December 2025.