CHAP. XX] PBOPEBTIES OF THE DIGITS OF NUMBERS. 465 Several86" gave §-n\+n-\-\= 1.. .1 forn^9, with generalization to any base. E. J. Moulton87 found the number of positive integers with r-f-1 digits fewer than p of which are unity (or zero). L. O'Shaughnessy88 found the number of positive integers < 10e which contain the digit 9 exactly r times. Books89 on mathematical recreations may be consulted. F. A. Halliday90 considered numbers N formed by annexing the digits of B to the right of A, such that N= (A+B)2, as for 81= (8+1)2. Set N=A'lQn+B. Then A(10n-'l) = (A+B)(A+.B-1), so that it is a question of the factors of 10n—1. *J. J. Osana91 discussed numbers of two and three digits. E. Gelin92 listed 450 problems, many being on digits. ^L'intermSdiaire des math., 25, 1918, 44-5. 87Amer. Math. Mpnthly, 24, 1917, 340-1. *Wd., 25, 1918, 27. 89E. Lucas, ArithmStique amusante, 1895. E. Fourrey, Relations Arithme'tiques, 1899. W. F. White, Scrap-Book of Elem. Math., etc. 90Math. Quest, and Solutions, 3, 1917, 70-3. wRevisfca Soc. Mat. Espafiola, 5, 1916, 156-160. «Mathesis, (2), 6, 1896, Suppl. of 34 pp.