458 History of the Theory of Numbers. [Chap, xx 
L. Gegenbauer^^'' proved generalizations of Cantor's^- theorems, allowing 
negative coefficients. Given the distinct positive integers ai, a2, . . ., every 
positive integer is representable in a single way as a linear homogeneous 
function of ai, a2, . . . with integral coefficients if each a^ is di\isible by a^-i 
and the quotient equals the number of permissible values of the coefficients 
of the smaller of the two. 
R. S. .Aiyar and G. G. Storr^^*" found the number p„ of integers the sum 
of whose digits (each >0) is n, by use of Pn= p„_i+ . . . +p„_9. 
E. Strauss''^ proved that, if ai, ao, . . . are any integers > 1, every positive 
rational or irrational number < 1 can be written in the form 
— f- 1 r... (ai<ai, a2<a2, •■ •), 
the a's being integers, and in a single way except in the case in which all the 
a„ beginning with a certain one, have their maximum values, when also a 
finite representation exists. 
E. Lucas^ noted that the only numbers having the same final ten digits 
as their squares are those ending with ten zeros, nine zeros followed by 1, 
8212890625 and 1787109376. He gave (ex. 4) the possible final nine digits* 
of numbers whose squares end with 224406889. He gave (p. 45, exs. 2, 3) 
all the numbers of ten digits to base 6 or 12 whose squares end with the same 
ten digits. Similar special problems were proposed by Escott and Palm- 
strom in Tlntermediaire des Mathematiciens, 1896, 1897. 
J. Kraus^ discussed the relations between the digits of a number 
expressed to two different bases. 
A. Cunningham**" called N an agreeable number of the mth order and 
nth degree in the r-ary scale if the m digits at the right of 'N are the 
same as the m digits at the right of A^" when each is expressed to base r; 
and tabulated all agreeable numbers to the fifth order and in some cases 
to the tenth. A number A of m digits is completely agreeable if the agree- 
ment of A with its nth power extends throughout its m digits, the condition 
being A"=A (mod r'"). 
E. H. Johnson^'' noted that, if a and r — \ are relatively prime and 
aa. . a (to r — 1 digits to base r) is divided by r — 1 , there appear in the 
quotient all the digits 1, 2, . . ., r— 1 except one, which can be found by 
dividing the sum of its digits by r — 1. 
C. A. Laisant^" stated that, if A = 123. . .n, written to base n+1, be 
multiplied by any integer <n and prime to n, the product has the digits 
of A permuted. 
^"'Sitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 618-27. 
'x^Math. Quest. Educ. Times, 47, 1887, 64. »*Acta Math., 11, 1887-8, 13-18. 
»'Th6orie des nombres, 1891, p. 38. Cf. Math. Quest. Educ. Times, (2), 6, 1904, 71-2. 
*Same by Kraitchik, Sphinx-Oedipe, 6, 1911, 141. 
"Zeitschr. Math. Phys., 37, 1892, 321-339; 39, 1894, 11-37. 
»*'British Assoc. Report, 1893, 699. »♦* Annals of Math., 8, 1893^, 160-2. 
»*«L'interm6diaire des math., 1894, 236; 1895, 262. Proof by "Nauticus," Mathesis, (2), 5, 
1895, 37-42. 
