Chap. XX] PROPERTIES OF THE DiGITS OF NUMBERS. 459 
Tables of primes to the base 2 are cited under Suchanek^" of Ch. XIII. 
There is a eollection^^'^ of eleven problems relating to digits. 
To find^*' the number <90 which a person has in mind, ask him to 
annex a declared digit and to tell the remainder on division by 3, etc. 
T. Hayashi^^ gave relations between numbers to the base r: 
123. . . {r-l}-(r-l)+r = l . . .1 (to r digits), 
{r-ljjr-2) ...321-(r-l)-l = {r-2} jr-2) . . . (to r digits). 
Several writers^*^ proved that 
123. . . {r-l}-(r-2)+r-l = {r-l} . . .321. 
T. Hayashi" noted that if A = 10+r(10)Hr2(10)^+ ... be multiplied or 
divided by any number, the digits of each period of A are permuted cyclically. 
A. L. Andreini^^" found pairs of numbers N and p (as 37 and 3) such that 
the products of N by all multiples ^ (J5 — l)p of p are composed of p equal 
digits to the base 5^ 12, whose sum equals the multiplier. 
P. de Sanctis^^ gave theorems on the product of the significant digits of, 
or the sum of, all numbers of n digits to a general base, or the numbers 
beginning with given digits or with certain digits fixed, or those of other 
types. 
A. Palmstrom^^ treated the problem to find all numbers with the same 
final n digits as their squares. Two such numbers ending in 5 and 6, 
respectively, have the sum 10"+ 1. If the problem is solved for n digits, 
the (n+l)th digit can be found by recursion formulae. There is a unique 
solution if the final digit (0, 1, 5 or 6) is given. 
A. Hauke^° discussed obscurely x'^^x (mod s'') for x with r digits to base 
s. If m = 2, while r and s are arbitrary, there are 2" solutions, v being the 
number of distinct prime factors of s. 
G. Valentin and A. Palmstrom^^ discussed x'^^x (mod 10"), for k = 2, 3, 
4, 5. 
G. Wertheim^^ determined the numbers with seven or fewer digits whose 
squares end with the same digits as the numbers, and treated simple prob- 
lems about numbers of three digits with prescribed endings when written to 
two bases. 
"'iSammlung der Aufgaben . . . Zeitschr. Math. Naturw. Unterricht, 1898, 35-6. 
«^«Math. Quest. Educ. Times, 6.3, 1895, 92-3. 
«Jour. of the Physics School in Tokio, 5, 1896, 153-6, 266-7; Abhand. Geschichte der Math. 
Wiss., 28, 1910, 18-20. 
»Jour. of the Physics School in Tokio, 5, 1896, 82, 99-103; Abhand., 16-18. 
"Ibid., 6, 1897, 148-9; Abhand., 21. 
"oPeriodico di Mat., 14, 1898-9, 243-8. 
»8Atti Accad. Pont. Nuovi Lincei, 52, 1899, 58-62; 53, 1900, 57-66; 54, 1901, 18-28; Memorie 
Accad. Pont. Nuovi Lincei, 19, 1902, 283-300; 26, 1908, 97-107; 27, 1909, 9-23; 28, 1910, 
17-31. 
''Skrifter udgivne af Videnskabs, Kristiania, 1900, No. 3, 16 pp. 
"Archiv Math. Phys., (2), 17, 1900, 156-9. 
^iForhandlinger Videnskabs, Kristiania, 1900-1, 3-9, 9-13. 
^^Anfangsgninde der Zahlenlehre, 1902, 151-3. 
