462 History of the Theory of Numbers. [Chap, xx 
W. Janichen^"'* 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 /j, as in Ch. XIX, 
s.W,.6) = 
0. 
E. N. Barisien^"'" noted that the sum of all numbers of n digits formed 
with p distinct digits f^O, of sum s, is 
s(p + 1)"-Mp(10"-^-1)/9 + (p + 1)10"-^). 
A. G^rardin^"'' listed all the 124 squares formed of 7 distinct digits. 
Several writers^^ treated the problem to find four consecutive numbers 
a^ 6 = a + l, c = a+2, d = a-\-3, such that (a)i = ll . . .1 (to a digits) is divisible 
by a + 1, (6)i by 26 + 1, (c)i by 3c+l, (d), by 4d+l. 
A. Cunningham and E. B. Escott^- 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. Barisien^^ noted that the squares of 625, 9376, 8212890625 end 
with the same digits, respectively. R. Vercellin^^ treated the same topic. 
E. Nannei^^ discussed a problem by E. N. Barisien: 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 writers^^ examined numbers of 6 digits which become divisible 
by 7 after a suitable permutation of the digits ; also^^ couples of numbers, 
as 18 and 36, 36 and 54, whose g. c. d. 18 is the sum of their digits. 
E. N. Barisien^^ gave ten squares not changed by reversing the digits, 
as676 = 26^ 
A. Witting^^ 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, 
102-402 = 201 -204, 213-936 = 312-639, 213-624 = 312-426. 
A. Cunningham^" treated three numbers L, M, N of I, m, n digits, 
respectively, such that N = LM, and A^ 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. 
'ocSphinx-Oedipc, 1907-8, 84-86. For p = n. Math. Quest. Educ. Times, 72, 1900, 126-8. 
•tx^/bid., 1908-9, 84-5. 
"L'intermddiaire des math., 16, 1909, 219; 17, 1910, 71, 203, 228, 286 [136]. 
"Math. Quest. Educat. Times, (2), 1.5, 1909, 27-8, 93-4. 
"Suppl.alPeriodicodiMat., 13, 1909,20-21. "Suppl.alPeriodicodiMat., 14, 1910-11, 17-20. 
"/bid., 13, 1909, 84-88. ^«L'interm6diaire 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. 
