History Of The Theory Of Numbers - I

462 HISTORY OF THE THEOEY OP NUMBERS. [CHAP, xx
W. Janichen506 stated that, if qp(x) denotes the sum of the digits of x to the base p and if p is a prime divisor of n, then, for /* as in Ch. XIX,
E. N. Barisien50c noted that the sum of all numbers of n digits formed with p distinct digits 5^0, of sum s, is
A. G6rardin50d listed all the 124 squares formed of 7 distinct digits.
Several writers51 treated the problem to find four consecutive numbers a, b = a+l, c = a+2, d = a+3, such that (a)i = ll . . .1 (to a digits) is divisible by o+l, (5)i by 2b+l, (c)i by 3c+l, (e^ by 4d+l.
A. Cunningham and E. B. Escott52 treated the problems to find integers whose squares end with the same n digits or all with n given digits; to find numbers having common factors with the numbers obtained by permuting the digits cyclically, as
259 = 7-37, 592 = 16-37, 925 = 25-37.
E. N. Barisien58 noted that the squares of 625, 9376, 8212890625 end with the same digits, respectively. R. Vercellin64 treated the same topic.
E. Nannei55 discussed a problem by E. N. Barlsien: Take a number of six digits, reverse the digits and subtract; to the difference add the number with its digits reversed; we obtain one of 13 numbers 0, 9900, . . . , 1099989. The problem is to find which numbers of six digits leads to a particular one of these 13, and to generalize to n digits.
Several writers56 examined numbers of 6 digits which become divisible by 7 after a suitable permutation of the digits; also57 couples of numbers, as 18 and 36, 36 and 54, whose g. c. d. 18 is the sum of their digits.
E. N. Barisien68 gave ten squares not changed by reversing the digits, as676=262.
A. Witting59 noted that, besides the evident ones 11 and 22, the only numbers of two digits whose squares are derived from the squares of the numbers with the digits interchanged by reversing the digits are 12 and 13. Similarly for the squares of 102 and 201, etc. Also,
102402 = 201-204, 213-936 = 312-639, 213-624 = 312-426.
A. Cunningham60 treated three numbers L, M, N of Z, m, n digits, respectively, such that N = LM, and N has all the digits of L and M and no others.
»°*Archiv Math. Phys., (3), 13, 1908, 361. Proof by G. Szego, 24, 1916, 85-6.
BOcSphinx-Oedipe, 1907-8, 84-86. For p = n, Math. Quest. Educ. Times, 72, 1900, 126-8.
«*/Wd., 1908-9, 84-5.
"L'interme'diaire des math., 16, 1909, 219; 17, 1910, 71, 203, 228, 286 [136].
B2Math. Quest. Educat. Times, (2), 15, 1909, 27-8, 93^1.
"Suppl. alPeriodicodi Mat., 13, 1909, 20-21. "Suppl. al Periodico di Mat., 14, 1910-11, 17-20.
KIbid.t 13, 1909, 84-88. "L'intermediaire des math., 17, 1910, 122, 214-6, 233-5.
"Ibid., 170, 261-4; 18, 1911, 207. "Mathesis, (3), 10, 1910, 65.
"Zeitschrift Math.-Naturw. Unterricht, 41, 1910, 45-50.
"Math. Quest. Educat. Times, (2), 18, 1910, 23-24.