460 History of the Theory of Numbers. [Chap, xx 
C. L. Bouton^ discussed the game nim by means of congruences between 
sums of digits of numbers to base 2. 
H. Piccioli^^" employed N = ai. . .a„ of n^3 digits and numbers a,^. . .a«„ 
and Qj^ . . . aj„ obtained from N by an even and odd number of transpositions 
of digits. Then 2o,, . . . a«„ = laj^ . . . aj^, 
W^^ a number of n digits to base R has r fixed digits, including the first, 
and the sum of these r is = — a (mod R — l), the number of ways of choosing 
the remaining digits so that the resulting number shall be divisible by R — l 
is the number of integers of n — r or fewer digits whose sum is = a (mod R — l) 
and hence is A^'+l or N, according as a=0 or a>0, where N= (/2"~'' — 1) 
G. Metcalfe^^'' noted that 19 and 28 are the only integers which exceed 
by unity 9 times the integral parts of their cube roots. 
A. Tagiuri^ proved that every number prime to the base g divides a 
number 1 ... 1 to base g (generalization of Plateau's^^ theorem) . 
If"^" A, B, C have 2, 3, 4 digits respectively and A becomes A' on re- 
versmg its digits, and 2A-1 = A', SB-2A + 10=B', 4C-B-\-l + [B/10] 
= C, then A = 37, B= 329, C= 2118. 
P. F. Teilhet^^ proved that we can form any assigned number of sets, 
each including any assigned number of consecutive integers, such that with 
the digits of the ^th power of any one of these integers we can form an 
infinitude of different qth. powers, provided q<m, where m is any given 
integer. 
L. E. Dickson^*" determined all pairs of numbers of five digits such that 
their ten digits form a permutation of 0, 1,. . ., 9 and such that the sum 
of the two numbers is 93951. 
A. Cunningham^^'' found cases of a number expressible to two bases by a 
single digit repeated three or more times. He^^" noted that all 10 digits or 
all >0 occur in the square of 10101010101010101 or of 1 . . .1 (to 9 digits), 
each square being unaltered on reversing its digits. 
He^^*^ and T. Wiggins expressed each integer ^ 140 by use of four nines, 
as 13 = 9+ V9+9/9, allowing also .9 = 1, ( V9) !, and the exponent V9, and 
cited a like table using four fours. 
j£45c ^= I (mod q), 1 ... 1 (with q^ digits to base r) is divisible by q"". 
W^^ the square of a number n of r digits ends with those r digits, then 
10''+1 — n has the same property. Also, {n — iy ends with the same r digits 
«Annals of Math., (2), 3, 1901-2, 35-9. GeneraUzed by E. H. Moore, 11, 1910, 93-4. 
^^'Nouv. Ann. Math., (4), 2, 1902, 46-7. 
"''Math. Quest. Educ. Times, (2), 1, 1902, 119-120. 
*'^Ibid, 63-4. 
"Periodico di Mat., 18, 1903, 45. 
«*<»Math. Quest. Educ. Times, (2), 5, 1904, 82-3. 
**L'interm6diaire dea math., il, 1904, 14-6. 
""Amer. Math. Monthly, 12, 1905, 94-5. 
"''Math. Quest. Educ. Times, (2), 8, 1905, 78. 
*^Ibid, 10, 1906, 20. «<iMath. Quest. Educ. Times, 7, 1905, 43-46. 
««7Wd., 7, 1905, 49-50. *^flbid., 7, 1905, 60-61. 
