Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 
45 
overlapping. Since p = x — l is to be prime, x is even (since a; = 3 makes z 
divisible by p = 2) . Hence x = 2P,z:fz = QiP:llP—S. By the theorem in (5) , 
QP and IIP — 3 have a common factor 2 or 3, so that P is either odd or divis- 
ible by 6. For P = Ql, the ratio is that of 126 to 22Z — 1 , which as before must 
have the common factor 3, whence l = 3t-\-l. Then z:fz = 4:{St-{-l) :22t-\-7, 
a ratio of relatively prime numbers, whence 22t+7'^f 4(31+1), and 
hence t = 2k, k = or k>3. For A; = 0, we obtain the pair 220, 284. 
The next value >3 of A; for which p = x — l and q = Qx—l are primes is 
k = Q, giving p = 443, g = 2663, numbers much larger than those in the 
(unnecessary) cases treated by Euler. Then z:jz = 4-37:271; set z = S7''d, 
d not divisible by 37; the cases e = 1, 2, 3 are excluded by the theorem in (5). 
For the remaining case P odd, P = 2Q-\-l, Euler treated those values ^100 
of Q, and also Q = 244, for which p and q are primes and obtained the pair 
in (I3), two pairs in (I5), and (14), (15). 
(53) Euler treated in §§112-7 various sets a, b, and obtained (a) and nine 
new pairs given in the table. 
In the following table of the 64 pairs of amicable numbers obtained by 
Euler, the numbering of any pair is the same as in Euler's list, but the pairs 
have been rearranged so that it becomes easy to decide if any proposed 
pair is one of Euler's. As noted by F. Rudio,^^^'' (37) contained the mis- 
print 3^ for 3^, w^hile (7) and (34) are erroneous, 220499 being composite 
(311-709); he checked that all other entries are correct. 
(4) 22-23{^27^^ 
(9) 2M3.17{3||09 
(47) 23/11-29-239 
^*'^ "^ 1191449 
^4»; ^ \29-47-59 
(21) 2417'5119 
(^)2^{83lo39(f-l««) 
(18) 2^{i:?^ 
(27) o«/79- 11087 
^^^^ ^ \383-2309 
(37) s^-sgi'ig'' 
(15) 32.7-13-4M63< 
(35) 32.5.19{7^227 
5-977 
5867 
(38) 2-5{7: 
(a)22 
60659 
23-29-673 
5-131 
17-43 
^•' ^ 1647-719 
(43) 2' 
(60) 
11-59-173 
47-2609 
2^-19-41 
26-199 
(oo\ 94/17-10303 
(2) 2^ 
[23-47 
\1151 
(19) 2^{i:t2? 
(25) 2»P-12671 
K^Q) z ^227-2111 
,3. 27/191-383 
(5) 32-7-13[ 
(14) 32-72-13-97 
5-17 
107 
5193 
1163 
(8) 3^-5-7{i?2S 
251 
107 
(1) 22{^jll 
i3) 2^(^32 
,45. 23(ll-23-1871 
V4^) ^ \467-1151 
.4Q. 23/11-163-191 
^4u; z 131.11807 
,r.,. /23-41-467 
^^^1 \25-19-233 
(23) 2^{}9gl439 
(50) 2423-47-9767 
\tM) z |i583.7io3 
,17s 24/23-1367 
^^'^ "^153-607 
(26) 2^{^|JJ^f 
/90X r,8/383-9203 
K^-6) z |ii5i.3067 
(7) 32-72-13{|5Y 
(10) 32-5- 19-37 (710^ 
(6) 3^-5-13gl9l9 
(51) 2^^^^ 
•131187 
-2267 
(29) 2^-ll{17:263 
,,,. 23/11-23-2543 
^**^ ^ 1383-1907 
(16) 2^(M:^^ 
(49) 24/17-167-13679 
KV6) z I809.51071 
(36) 2^-67{|72411 
.24) 26/59-1103 
(.Z4J L J79.827 
{b2)Z^-l-\Z^^lll]f 
'"^Bibliotheca Math., (3), 14, 1915, 351-4. 
