CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 45 overlapping. Since p=x — 1 is to be prime, x is even (since x=3 makes z divisible by p = 2). Hence£=2P,3:Jz=6P:llP-3. By the theorem in (5), 6P and 1 IP - 3 have a common factor 2 or 3, so that P is either odd or divisible by 6. For P = 6Z, the ratio is that of 126 to 221-1, which as before must have the common factor 3, whence Z=3H-L Then z:fz=*4(3t+l):22J+7, a ratio of relatively prune numbers, whence 22J+7^j4(3£+l), and hence t=2k, fc = 0 or &>3. For fc=0, we obtain the pair 220, 284. The next value >3 of k for which p=x—l and g=6x—1 are primes is k = 6, giving p=443, #=2663, numbers much larger than those in the (unnecessary) cases treated by Euler. Then 2:J«=4-37:271; set z=37ed, 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-f 1, Euler treated those values ^ 100 of Qj and also Q=244, for which p and q are primes and obtained the pair in (13), two pairs in (15), and (14), (15). (53) Euler treated in §§112-7 various sets a, 6, 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,3646 (37) contained the misprint 33 for 32, while (7) and (34) are erroneous, 220499 being composite (311-709); he checked that all other entries are correct. 7-19-107 7-60659 /«i (51) 5-13-1187 (4) (9) fl 1-29-239 117-4799 » 245'131 1 2 \1743 ,x 23/11-23-1619 J) 2 1647-719 ,v 03/11-59473 5J z 147-2609 /.,v (45) 113-107 ,3/11.23-1871 , 03/1M63-191 1 ^ 131-11807 v /28-41-467 ' 12«-19-233 k 9,/ll-23-2543 1 ^ 1383-1907 > oj/17-79 ) 2 123-59 , 04/17-16743679 ' z 1809-51071 04/17-) 2 \239 17-5119 239-383 I 04/23-467 1 L 1103-107 k 04/47-89 , oe/79-11087 1 ^ 1383-2309 k Q2,/7-11-29 ^ 04/17-10303 -; ^ 1167-1103 1 24{n54i7 y, 04/23-479 *> 2 189-127 .x 2J37-12671 ^ z \227-2111 , 27/191-383 1 L \73727 \149-191 /23-47-9767 J23-1367 J53-10559 (36) 2«/59'1103 2 |79.827 -9203 (7) (14) 32 (8) 3«-J (6) 3M&Bibliotheca Math., (3), 14, 1915, 351-4.