58 HlSTOBY OF THE THEOEY OF NUMBERS. [CHAP. II
G6rardin68 gave five new solutions of (i):
x=3.11.31.443.499, t/ = 29.3.54.13.37.61.157.
z=2.3331.443.449, y=273.5411.13.37.61.157.
2=11.17.41.43.239.307.443.499,
i/=212.33.57.7.11.133.292.37.61.157. x=2.11.l7.23.41.211.467.577.853,
2/ = 210.34.53.7.132.17.292.53.61.113.193.197. z=3311.13.23.83.193.701,
t/ = 2933537.11.13.17.53.61.97.149, 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.
PRODUCT OF ALIQUOT PARTS.
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