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458                        HlSTOKY OF THE THEORY OF NUMBERS.               [CHAP. XX
L. Gegenbauer316 proved generalizations of Cantor's12 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 ax is divisible by ax_j and the quotient equals the number of permissible values of the coefficients of the smaller of the two.
R. S. Aiyar and G. G. Storr31' found the number pn of integers the sum of whose digits (each >0) is n, by use of pn— pn_i+ . . . +pn_g.
E. Strauss32 proved that, if a1? o2, . . . are any integers > 1, every positive rational or irrational number < 1 can be written in the form
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. Lucas33 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 l'IntermŁdiaire des Math&naticiens, 1896, 1897.
J. Kraus34 discussed the relations between the digits of a number expressed to two different bases.
A. Cunningham340 called AT 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 Nn 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 N of m digits is completely agreeable if the agreement of N with its nth power extends throughout its m digits, the condition being Nn=N (mod rm).
E. H. Johnson346 noted that, if a and r — 1 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. Laisant340 stated that, if N = 123. . .n, written to base w+1, be multiplied by any integer <n and prime to n, the product has the digits of N permuted.
«&Sitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 618-27.
»lcMath. Quest. Educ. Times, 47, 1887, 64.          "Acta Math., 11, 1887-8, 13-18.
MTh6orie 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, 1893-4, 160-2.
«*L'interme*diaire des math., 1894, 236; 1895, 262.    Proof by "Nauticus," Matheais, (2), 5,
tone   OT  Aft