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58                         HlSTOBY OF THE THEOEY OF NUMBERS.                    [CHAP. II
G6rardin68 gave five new solutions of (i):
x=,                 t/ =
z=2.3331.443.449,                   y=273.5411.
i/= x=2.11.l7.,
2/ = z=3311.,
t/ = 2933537., the last following from his67 fourth pair in view of
cr(39ll3): (r(2333)==283.112612: 233.52 = 22112612: 52.
A. Cunningham and J. Blaikie69 found solutions of the form x=2"p of s(x) =02, where s(n) is the sum of the divisors <n of n.
Paul Halcke75 noted that the product of the aliquot parts of 12, 20, or 45 is the square of the number; the product for 24 or 40 is the cube; the product for 48, 80 or 405 is the biquadrate.
E. Lionnet76 defined a perfect number of the second kind to be a number equal to the product of its aliquot parts. The only ones are p3 and pq, where p and q are distinct primes.
"L'mtermeMiaire des math., 24, 1917,132-3.
"Math. Quest. Educ-. Times, (2), 7, 1905, 68-9.
"Deliciae Math, oder Math. Sinnen-Confect, Hamburg, 1719, 197, Exs. 150-2.
78Nouv. Ann. Math., (2), 18,1879, 306-8.   Lucas, Th6orie des nombres, 1891, 373, Ex. 6