454 History of the Theory of Numbers. [Chap. 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. 
Several^" 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 writer^ proposed the problem to find a number n given 
the product of n by the number obtained from n by writing its digits in 
reverse order [Laisant*^]. 
P. T^denat^ 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 a^ = 10'*~^64-a, we can find a digit A to annex at the 
left of a to obtain a desired number 10"~^A + a of n digits. Squaring the 
latter, we obtain the condition {2a — l)A= —b (mod 10). 
J. F. Frangais^ noted the solutions 
x = 2"p = 5"5 + l, x2 = 10"p5+a:, 
2/ = 5«r = 2"s + l, y^ = lO"rs-\-y, 
in which the resulting condition 2"p — 5"^ = 1 or 5"r — 2"s= 1 is to be satisfied. 
Special solutions are given by n = 1, p = 3; n = 2, p = 19; n = 3, p = 47; n = 4, 
p = 586; etc., to n = 7. 
J. D. Gergonne^ generalized the problem to base B. Then 
x(rc-l)=5"t/. 
Let p, q be relatively prime and set jB" = pq. Then x = pt, x — l=qu, or vice 
versa. The condition pt — qu = l is solved for t, u. When B = 10, n = 20, 
the least u is 81199. 
Anonymous writers^ stated and proved by use of the decimal fraction for 
1/n that every number divides a number of the form 9 ... 90 ... 0. 
A. L. Crelle^ 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. 
Several^" found a square whose root has two digits, their quotient be- 
ing equal to their difference. By x/y=x—y, x=i/+l + l/(2/ — 1), an inte- 
ger, whence y=2, x=4t. Thus the squares are 24^ or 42^. 
The^^ three digits of a number are in geometrical progression; the prod- 
uct 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 
^ffLadies' Diary, 1811-12, Quest. 12.31; 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. 
^Ibid., 322-7. 
*Ibid., 19, 1828-9, 256; 20, 1829-30, 304-5. 
Ubid., 20, 1829-30, 349-3.52; Jour, fiir Math., 5, 1830, 296. 
^"Ladies' Diary, 1820, 36, Quest. 1347. 
'I'Ibid., 1822, 33, Quest. 1374. 
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