48 HlSTOBY OF THE THBOKY OF NUMBERS. [HAP. I P. Seelhoff373 treated Euler's364 problems 1 and 2 by Euler's methods (though the contrary is implied), and gave about 20 pairs of amicable numbers due to Euler, with due credit for only three pairs. The only new pairs (pp. 79, 84, 89) are .27210 iq oQ/83.1931 26fl39.863 6 t Aa-iy-^162287 [167.719. E. Catalan374 stated empirically 'that if nl is the sum of the divisors <n of n, and n2 is the sum of the divisors <Ui of rii, etc., then n, %, n2, . . . have a limit X, where X is unity or a perfect number. J. Perrott376 [Perott] noted that there is no limit for n = 220, since ni = n3= . . . =284, 7*2 = ^4 = . . . =220. H. LeLasseur376 found that for n< 35 the numbers (1) are all odd primes, and hence give amicable numbers, only when n=2y 4, 7. Josef Bezdicek377 gave a translation into Bohemian of Euler,364 without credit to Euler, and a table of 65 pairs of amicable numbers. Aug. Haas378 proved that, if M and N are amicable numbers, where m and n range over all divisors of M and JV, respectively. For, Sm=Sn=J!f+#, so that yl__2m_M+N ^l_2n_M+N m-M M~' n~N W~' If M-N, N is perfect and the result becomes that of Catalan.145 A. Cunningham379 considered the sum s(n) of the divisors <n of n and wrote s2(n) for s-js(n)f, etc. For most numbers, sk(ri) = 1 when k is sufficiently large. There is a small class of perfect and amicable numbers, and a small class of numbers n (even when n<1000) for which sk(n) increases beyond the practical power of calculation [cf. Catalan374]. A. G6rardin380 proved that the only pairs 22-5x, 2?yz of amicable numbers, where x, y} z are odd primes, are Euler's (a), (ft) ; the only pairs 24-23z, 2V are Euler's (17), (19), (20). He cited the Exercices d'arithm6tique of Fitz-Patrick and Chevrel; also Dupuis' Table de logarithmes, which gives 24 pairs of amicable numbers. Gfrardin381 proved that the only pair 8xy, 322 is Euler's (60) . He made an incomplete examination of 16-53s, 162/2, but found no new pairs. »7»Archiv Math. Phya., 70, 1884, 75-89. »7«Bull. Soc. Math. France, 16, 1887-8, 129. Mathesis, 8, 1888, 130. »76Jbid., 17, 1888-9, 155-6. 87eLucas, Thgorie des nombres, 1, 1891, 381. 377Casopis mat. a fys., Praze (Prag), 25, 1896, 129-142, 209-221. 378J6id., 349-350. "Proc. London Math. Soc., 35, 1902-3, 40. "°Mathesis, 6, 1906, 41-44. '"Sphinx-Oedipe, Nancy, 1906-7, 14-15, 53.