Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 49 
G^rardin^^^ proved that the three numbers (1) with n = m-f 2 are not all 
primes if 34< m^ 60, the cases m = 38 and 53 not being decided. Replacing 
m by m+1 and A; by 2^+1 in case (li) of Euler^^'*, we get the pair 2"pg, 
2"r, where n = m+2g-]-2, 
p = 2"*+2''+^P-l, 5 = 2"*+^P-l, ^ = 22'"+2''+3p2_i^ 
with P = 2^^'*"^ + l. For 9^ = 0, we have the case (1) just mentioned; all 
values m^200 are excluded except m = 38, 74, 98, 146, 149, 182, 185, 197. 
The case gr= 1 is excluded since i/ or 2 is a difference of two squares. For 
g = 2, all values m ^ 60 are excluded except m= 29, 34, 37, 49. For g = 3, all 
values <100 are excluded except m = 8, 15, 23, 92. 
0. Meissner,^^^ using the notation of Cunningham,^^^ noted that n and 
s{n) are amicable if s^(n)=n and raised the question of the existence of 
numbers n for which s''{n)—n for k^S, so that n, s{n),. . .,s*~^(n) would 
give amicable numbers of higher order. He asked if the repetition of the 
operation s, a finite number (k) of times always leads to a prime, a perfect 
or amicable number; also if k increases with n to infinity. On these ques- 
tions, see Dickson^^^ and Poulet.^^^ 
A. Gerardin^^^ stated that the only values n<200 for which the 
numbers (1) are all primes are the three known to Descartes. 
L. E. Dickson^^^ obtained the two new pairs of amicable numbers 
2*-12959-50231, 2*- 17- 137-262079; 2*- 10103-735263, 2^-17-137-2990783, 
by treating the type IQpq, 16-17-137r, where p, q, r are distinct odd primes. 
These are amicable if and only if 
p = m+9935, g = w+9935, r = 4(w+n) +88799, wn = 2^3*7-23-73. 
Although Euler^^ mentioned this type (33) in §95, he made no discussion 
of it since r always exceeds the limit 100000 of the table of primes accessible 
to him. An examination of the 120 distinct cases led only to the above 
two amicable pairs. 
Dickson^^® proved that there exist only five pairs of amicable numbers 
in which the smaller number is <6233, viz., (1), (a), (^), (60) in Euler's^^* 
table, and Paganini's^^^ pair. In the notation of Cunningham,^^^ the chain 
n, s{n), s^{n), . . .is said to be of period k if s^(n) =n. The empirical theorem 
of Catalan'^* is stated in the corrected form that every non-periodic 
chain contains a prime and verified for a wide range of values of n. In 
particular, if n<6233, there is no chain of period 3, 4, 5, or 6. For k odd 
and > 1, there is no chain arii, an2, . . . , aUk of period k in which /ii, . . . , n^ 
have no common factor and each rij is prime to a> 1. 
'^^Sphinx-Oedipe, 1907-8, 49-56, 65-71 ; some details are inaccurate, but the results correct. 
'S'Archiv Math. Phys., (3), 12, 1907, 199; Math.-Naturw. Blatter, 4, 1907, 86 (for k=3). 
'** Assoc, frang. avanc. sc, 37, 1908, 36-48; I'intenn^diaire des math., 1909, 104. 
'8*Amer. Math. Monthly, 18, 1911, 109. 
"•Quart. Jour. Math., 44, 1913, 264-296. 
