58 History of the Theory of Numbers. [Chap.ii 
G^rardin^^ gave five new solutions of (i) : 
X = 3.11.31.443.499, i/ = 2^3.5M3.37.61.157. 
x = 2.3^31.443.449, 2/ = 2^3.5ni. 13.37.61.157. 
a; = 1 1 . 17.41 .43.239.307.443.499, 
2/ = 2^2 3^5'.7.11. 13^29^.37.61. 157. 
x = 2.11. 17.23.41.211.467.577.853, 
t/ = 2^''.3^5l7.13M7.292.53.61. 113.193.197. 
x = 3ni. 13.23.83.193.701, 
?/ = 293'537.11. 13.17.53.61.97.149, 
the last following from his*^^ fourth pair in \'iew of 
a{SnV): a{2'S') = 2'3.nm^: 233.52 = 2=11-612; 5=. 
A. Cunningham and J. Blaikie^^ found solutions of the form x = 2'p of 
s{x) =g^, where s{n) is the sura of the divisors <n of n. 
product of aliquot parts. 
Paul Halcke'^^ 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. Lionnet"^ defined a perfect number of the second kind to be a number 
equal to the product of its aliquot parts. The only ones are p^ and pq, 
where p and q are distinct primes. 
"L'interm^diaire des math., 24, 1917, 132-3. 
"Math. Quest. Educ. Times, (2), 7, 1905, 68-9. 
"Dehciae Math, oder Math. Sinnen-Confect, Hamburg, 1719, 197, Exs. 150-2. 
"Nouv. Ann. Math., (2), 18, 1879, 306-8. Lucas, Th6orie des nombres, 1891, 373, Ex. 6 
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