58 HlSTOBY OF THE THEOEY OF NUMBERS. [CHAP. II
G6rardin68 gave five new solutions of (i):
x=22.214.171.1243.499, t/ = 126.96.36.199.37.61.157.
2/ = 188.8.131.52.132.17.2184.108.40.206.193.197. z=33220.127.116.11.193.701,
t/ = 2933518.104.22.168.22.214.171.124, 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