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CHAP. XX]           PROPERTIES OF THE DIGITS OF NUMBERS.                   459
Tables of primes to the base 2 are cited under Suchanek80 of Ch. XIII. There is a collection34* of eleven problems relating to digits. To find346 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. Hayashi36 gave relations between numbers to the base r:
123... {r~l}-(r-l)+r = l.. .1 (to r digits),
{r-1} {r-2}.. .321-(r-l) -1 = {r-2} {r-2}... (to r digits).
Several writers36 proved that
123... {r-l}-(r-2)+r-l = {r-1}.. .321.
T. Hayashi37 noted that if A = 10+r(10)2+^(10)8+... be multiplied or divided by any number, the digits of each period of A are permuted cyclically.
A. L. Andreim"37a found pairs of numbers N and p (as 37 and 3) such that the products of N by all multiples g (J3 —l)p of p are composed of p equal digits to the base B^ 12, whose sum equals the multiplier.
P. de Sanctis38 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. Palmstrom39 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 10n+L If the problem is solved for n digits, the (n-fl)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. Hauke40 discussed obscurely xm=x (mod sr) 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. Palmstrom41 discussed xk=x (mod 10n), for A; = 2, 3, 4,5.
G. Wertheim42 determined the numbers with seven or fewer digits whose squares end with the same digits as the numbers, and treated simple problems about numbers of three digits with prescribed endings when written to two bases.
"dSammlung der Aufgaben... Zcitschr. Math. Naturw. Unterricht, 1898, 35-6.
MeMath. Quest. Educ. Times, 63, 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.
MJour. of the Physics School in Tokio, 5, 1896, 82, 99-103; Abhand., 16-18. "Ibid., 6, 1897, 148-9; Abhand., 21. »7«Periodico di Mat., 14, 1898-9, 243-8. "Atti 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,
"Skrifter udgivne af Videnskabs, Kristiania, 1900, No. 3, 16 pp. "Archiv Math. Phys., (2), 17, 1900, 156-9. "Forhandh'nger Videnskabs, Kristiania, 1900-1, 3-9, 9-13. ^Anfangflgrtinde der Zahlenlehre, 1902, 151-3.