460 HISTORY OF THE THEOKY OF NUMBEKS. [CHAP. XX C. L. Bouton" discussed the game nim by means of congruences between of of numbers to base 2. IL Piceioli4*1 employed AT = ai.. ,an of n^3 digits and numbers at-.. .o% and a,t, . . a,„ obtained from AT by an even and odd number of transpositions of Then Sais... ain == Sa^... a,n. Ifl3h a number of n digits to base E has r fixed digits, including the first, and the sum of these r is s -a (mod fi-1), the number of ways of choosing tin* remaining digits so that the resulting number shall be divisible by fi-1 m the? number of integers of n—r or fewer digits whose sum is =a (mod #—1) and hence in N + l or AT, according as a=0 or a>0, where ]V=(JRn-r~l) /(«-!). («. Metcalfe43* noted that 19 and 28 are the only integers which exceed by unity 9 times the integral parts of their cube roots. A. Tagiuri44 proved that every number prime to the base g divides a number 1. . .1 to base g (generalization of Plateau's18 theorem). If443 A, B, C have 2, 3, 4 digits respectively and A becomes A' on re-it* digits, and 2A-1 = A', 3B-2A+10=5', 4C-B+1+[B/10] ** f", then A ** 37, B- 329, C= 2118. 1*. F. Teilhet46 proved that we can form any assigned number of sets, including any assigned number of consecutive integers, such that with the digits of the qth power of any one of these integers we can form an infinitude of different gth powers, provided <?<w, where m is any given integer. L. K. IMckson450 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. C 'uimingham4*6 found cases of a number expressible to two bases by a winnle digit, repeated three or more times. He45c noted that all 10 digits or all >0 occur in the square of 10101010101010101 or of 1.. .1 (to 9 digits), each wquare being unaltered on reversing its digits. He4* and T. Wiggins expressed each integer <£ 140 by use of four nines, as Ki 9 f- \AH-9/9, allowing also .9 = 1, (*N/9)!, and the exponent V9, and riti»ii a like table lining four fours. If4*' r 1 (mod q), 1 ... 1 (with qn digits to base r) is divisible by qn. lfif>/ the Hquare of a number n of r digits ends with those r digits, then 11f i 1 - n hiiH the wime property. Also, (n-1)3 ends with the same r digits *».\imaln of Math., (2), .'i, lftOl-2, 35-9. Generalized by E. H. Moore, 11, 1910, 93-4. ""NOUV. Ann. Math., M), 2, 1902, 40-7. "'•Ninth. QucHt. Ivluc. TiincH, (2), 1, 1902, 119-120. **ltnd, tKt -1. *«IVn»MiJi-« di Mat., 18, HX)3, 45. *b»M»th. Ciui-Ht. Kduc. Timt-fl, (2), 5, 1904, 82-3. ttI/mt4T«i^(liftire <l«i math., 11, 1904, 14-6. ^Anwr. Math. Monthly, 12, 1905, 94-5. «*f'Mnth. Uuf»»t. K«luc. Timcw, (2), 8, 1905, 78. •"Ibitl. H), 1IHB, 20. *^Math. Quest. Educ. Times, 7, 1905, 43-46. "•lintl., 7' HK)f>. 4<>-f>0. "'Ibid., 7, 1905, 60-61.