History Of The Theory Of Numbers - I

50 HlSTOBY OP THE THEOKY OP NUMBEBS. [CHAP. I
P. Poulet387 discovered the chain of period five, n* 12496 =2M1-71, s(n)=24-19-47, s2(n) = 24-967, s3
$4(n)=23-1783,
with 56(n) =n; and noted that 14316 leads a chain of 28 terms. GENEBALIZATIONS OP AMICABLE NUMBEBS.
Daniel Schwenter82 noted in 1636 that 27 and 35 have the same sum of aliquot parts. Kraft363 noted in 1749 that this is true of the pairs 45, 3-29; 39, 55; 93, 145; and 45, 13-19. In 1823, Thomas Taylor102 called two such numbers imperfectly amicable, citing the pairs 27, 35; 39, 55; 65, 77; 51, 91; 95, 119; 69, 133; 115, 187; 87, 247. George Peacock400 used the same term.
E. B. Escott401 asked if there exist three or more numbers such that each equals the sum of the [aliquot] divisors of the others.
A. Ge"rardin402 called numbers with the same sum of aliquot parts nombresassocie*s, citing 6 and 25; 5-19, 7-17, and 11-13, and many more sets. An equivalent definition is that the n numbers be such that the product of n— 1 by the sum of the aliquot divisors of any one of them shall equal the sum of the aliquot divisors of the remaining n—1 numbers.
L. E. Dickson403 defined an amicable triple to be three numbers such that the sum of the aliquot divisors of each equals the sum of the remaining two numbers. After developing a theory analogous to that by Euler364 for amicable numbers, Dickson obtained eight sets of amicable triples in which two of the numbers are equal, and two triples of distinct numbers :
293-337a, 5-16561a, 99371a (a=25-3-13),
3-896, 11-296, 3596 (6 = 214.5-19-3M51).
""L'interm&iiaire des math., 25, 1918, 100-1. '"Encyclopaedia Metropolitana, London, I, 1845, 422. 401L'interm6diaire des math., 6, 1899, 152. 4WSphinx-Oedipe, 1907-8, 81-83. 40lAmer. Math. Monthly, 20, 1913, 84-92.