Chap. XX] PROPERTIES OF THE DiGITS OF NXH^BERS. 461 
as 71 — 1. If the cube of a number n of r digits ends with those r digits, 
W—n has the same property. 
P. Ziihlke^® proved the three theorems of Palmstrom^^ and gave all solu- 
tions of x^^x (mod 10^) for p = 3,. . ., 12. 
M. Koppe^^ noted that by prefixing a digit to a solution 0, 1, 5 or 6 
of x^=x (mod 10) we get solutions of a;^=a; (mod 10^), then for 10^, etc. We 
can pass from a solution with n digits for 10" to solutions with 2n digits 
for 10^". He treated also x^=x (mod 10"). 
G. Calvitti^^ treated the problem: Given a number A, a set C of 7 digits, 
and a number p prime to the base g, to find the least number x of times the 
set C must be repeated at the right of A to give a number Nx=A (mod p). 
The condition is G{Ni—No)=0 (mod p), where 
o^"^ — 1 
If iVi— iVo=0, any x is a solution. If not, the least value X of a: makes 
G=0 (mod p/p), where p is the g. c. d. of Ni — Nq and p. Then X is the 
1. c. m. of Xi, . . . , X;t, where X^ is the least root of G=0 (mod Pi), if p/p is the 
product of pi, . . . , Pk, relatively prime in pairs. Hence the problem reduces 
to the case of a power of a prime p. Write (a)^ for (a^ — l)/(a — 1). It is 
shown that the least root of (a)j.= (mod p'') is mp^~\ where m is the least 
root of (a)^=0 (mod p), and p' is the highest power of p dividing (a)^. 
Given any set C of digits and any number p prime to the base g, there exist 
an infinitude of numbers C . . .C divisible by p. 
A. G^rardin^^'' added 220 to the sum of its digits, repeated the operation 
18 times and obtained 418; 9 such operations on 284 gave 418. A. Boutin 
stated that if a and b lead finally to the same number, neither a nor 6 is 
divisible by 3, or both are divisible by 3 and not by 9, or both are divi- 
sible by 9. 
E. Malo^^ considered periodicity properties of A and a in 
5' = 10'"A„.p+ap (ap<10^ k=n-2'^-^+p, 0^p^2"'-^-l) , 
and solved Cesaro's^^ three problems on the digits of powers of 5. 
A. L. Andreini^° noted that the squares of A and B end with the same 
p digits if and only if the smaller of r+s and u-\-v equals p, where 
iH-J5 = a-2'^-5", A-B=^-T-5'' 
«Sitz. Berlin Math. Gesell., 4, 1905, 10-11 (Suppl., Archiv Math. Phys., (3), 8, 1905). 
"Ibid., 5, 1906, 74-8. (Suppl., Archiv, (3), 11, 1907.) 
"Periodico di Mat., 21, 1906, 130-142. 
""Sphinx-Oedipe, 1, 1906, 19, 47-8. Cf. I'interm^d. math., 22, 1915, 134, 215. 
*'Sur certaines propri6t6s arith. du tableau des puissances de 5, Sphinx-Oedipe, 1906-7, 97-107; 
reprinted, Nancy, 1907, 13 pp., and in Nouv. Ann. Math., (4), 7, 1907, 419-431. 
"II Pitagora, Palermo, 14, 1907-8, 39-^7. 
