454 HlSTOKY OF THE THEORY OP NUMBEES. [CnAP. XX three 4's, the least is 1444. J. Davey discussed only numbers of 3 or 4 digits of which the last 2 or 3 are equal, respectively. Several1* found that the squares 169 and 961 are composed of the same digits in reverse order, have roots of two digits in reverse order, while the sum of the digits in each square equals the square of the sum of the digits in each root; finally, the sum of the digits in each root equals the square of their difference. An anonymous writer2 proposed the problem to find a number n given the product of n by the number ob tamed from n by writing its digits in reverse order [Laisant18]. P. T6denat3 considered the problem to find a number of n digits whose square ends with the same n digits in the same order. If a is such a number of n— 1 digits, so that a2=10n~16+o, we can find a digit A to annex at the left of a to obtain a desired number 10n""1A+a of n digits. Squaring the latter, we obtain the condition (2a— l)J.s= — & (mod 10). J. F. Frangais4 noted the solutions hi which the resulting condition 2np— 5ng= 1 or 5nr — 2ns= 1 is to be satisfied. Special solutions are given by n=l, p = 3; n=2, p = 19; n = 3, p=47; n=4, p = 586; etc., ton =7. J. D. Gergonne5 generalized the problem to base B. Then Let p, q be relatively prime and set Bn = pq. Then x = pt, x — 1 = qu, or vice versa. The condition pt— qu~l is solved for t, u. When 5 = 10, n=20, the least u is 81199. Anonymous writers6 stated and proved by use of the decimal fraction for l/n that every number divides a number of the form 9 ... 90 ... 0. A. L. Crelle7 proved the generalization: Every number divides a number obtained by repeating any given set of digits and affixing a certain number of zeros, as 23... 230... 0. Several70 found a square whose root has two digits, their quotient being equal to their difference. By x/y=x—y, x=y+l+l/(y— 1), an integer, whence ?/= 2, x— 4. Thus the squares are 242 or 422. The7b three digits of a number are in geometrical progression; the product of the sum of their cubes by the cube of their sum is 1663129; if the number obtained by reversing the digit be divided by the middle digit, the ^Ladies' Diary, 1811-12, Quest. 1231; Leybourn, I. c., 153-4. 'Annales de Math, (ed., Gergonne), 3, 1812-3, 384. 'Ibid., 5, 1814-5, 309-321. Problem proposed on p. 220. *Ibid., 321-2. 8/6id., 322-7. •/bid., 19, 1828-9, 256; 20, 1829-30, 304-5. 7/6id., 20, 1829-30, 349-352; Jour, fur Math., 5, 1830, 296. '"Ladies' Diary, 1820, 36, Quest. 1347. bid., 1822, 33, Quest. 1374.