46 History of the Theory of Numbers. [Chap, i 
(31) S^-5'13{ll:]f (54) 3«.5^{}1:^9.179 ^g^ ^,_^_^^,^^l29.5m 
(33) 33.5.13.19^/2711 (32) 3^-5-13{^^:^J (12) S^.r-lVlsl^lf^^' 
(41)33.7.13.23|JJ;1%367 ^g^^ 3,.-,. ^g.^gjl 1-220499 ^^^^^^ 
(30) 33.5{[7^1j (55) ■S^-5{',lll%^ (42) 33.5.23(11 -.J^SIT 
(11) 3^.5.1l{29g89 (56) 3^.7.11M9{g97019 (57)3^.7.11M9{^3^6959^ 
(53) 3».7M3.53{ll4211 (53) 3s.7M3.19{4™19 (59^ 3^.7M3.19{53g6959^ 
Euler's final list of 61 pairs did not include the pairs a, jS, 7, although he 
had obtained a four times in the body of his paper, viz., in (2), (3i), (63); 
jS twice in (3i); 7 in (2). Moreover, these three unlisted pairs occur as 
VIII, IX, and XIII among the 30 pairs in Euler's^^^ earlier list, a fact noted 
on p. XXVI and p. LVIII of the Preface by P. H. Fuss and N. Fuss to 
Euler's Comm. Arith. Coll., who failed to observe that these three pairs 
occur in the text of Euler's present paper. Nor did these editors note 
that the fourth mentioned case of divergence between the two lists is due 
merely to the misprint^^^'' of 57 for 47 in (43) of the present list, so that 
the correctly printed pair XXVIII of the list of 30 is really this (43) and 
not a new pair, as supposed by them. 
From the fact that Euler obtained in his posthumous tract®^ on amicable 
numbers the pairs a, jS (once on p. 631 and again on p. 633 and finally on 
p. 635), the editors inferred, p. LXXXI of the Preface, that the tract differs 
in analysis from the long paper just discussed. But no new pairs are found, 
while the cases treated on pp. 631-2 are merely problems 1 and 2 of Euler's 
preceding paper. It is different with p. 634, where Euler started with two 
numbers like 71 and 5-11 which, by his table, have the same sum, 72, of 
divisors, and required a number a relatively prime to them such that 71a 
and 55o are amicable. The single condition is 72J a=(71+55)o, whence 
ja:a = 7A. Thus a has the factor 4. If a = 46, where b is odd, then 
Ch = h = l, and the pair 284, 220 results. The case a = 86 is impossible. This 
method was used in a special way by Kraft^^^ who limited the numbers 
from which one starts to a prime and a product of two primes. 
In the Encyclopedie Sc. Math., I, 3i, p. 59, note 320, it is stated that this 
posthumous tract contains four pairs not in Euler's list of 61, two pairs being 
those of Fermat^^^ and Descartes.^^^ But these were fisted as (2) and (3) 
by Euler and were obtained by him in case (li) and attributed to Descartes. 
E. Waring^^^ noted that 2'*x, 2''yz are amicable if 
2"'?/z_2"+^ + l 2^" 
2"-l ' i/-2"+l 
where x^ y, z are primes and ?/ — 2"H-1 divides 2^**. He cited the first two 
such pairs of amicable numbers. 
3"«G. Enestrom, Bibliotheca Math., (3), 9, 1909, 263. 
3«*Meditationes algebraicae, 1770, 201; ed. 3, 1782, 342-3. 
