5 
11 
23 
47 
2 
4 
8 
16 
6 
12 
24 
48 
71 
287 
1151 
40 History of the Theory of Numbers. [Chap, i 
Alkalacadi,^" a Spanish Arab (tl486), showed the method of finding the 
least amicable numbers 220, 284. 
Nicolas Chuquet^^ in 1484 and de la Roche^^ in 1538 cited the amicable 
numbers 220, 284, "de merueilleuse familiarite lung auec laultre." In 1553, 
Michael StifeP^ (folios 26v-27v) mentioned only this pair of amicable num- 
bers. The same is true of Cardan, ^^ of Peter Bungus^- (Mysticae numerorum 
signif., 1585, 105), and of TartagUa.^^ Reference may be made also to 
Schwenter." 
In 1634 Mersenne^"^ (p. 212) remarked that "220 and 284 can signify 
the perfect friendship of two persons since the sum of the aliquot parts of 
220 is 284 and conversely, as if these two numbers were only the same thing." 
According to Mersenne's^"^ statement in 1636, Fermat^^ found the 
second pair of amicable numbers 
17296 = 2'.23.47, 18416 = 2^-1151, 
and communicated to Mersenne^°^ the general rule: Begin with the geo- 
metric progression 2, 4, 8, ... , write the prod- 
ucts by 3 in the line below; subtract 1 from 
the products and enter in the top row. The 
bottom row is 6-12-1, 12-24-1,. . .When a 
mmiber of the last row is a prime (as 71) and 
the one (11) above it in the top row is a prime, 
and the one (5) preceding that is also a prime, then 71.4 = 284, 5-11-4 = 220 
are amicable. Similarly for 
1151-16 = 18416, 23-47-16 = 17296, 
and so to infinity. [The rule leads to the pair 2"/i<, 2'*s, where h, t, s are 
given by (1).] 
Descartes^^^ gave the rule: Take (2 or) any power of 2 such that its 
triple less 1, its sextuple less 1, and the 18-fold of its square less 1 are all 
primes;* the product of the last prime by the double of the assumed power 
of 2 is one of a pair of amicable numbers. Starting with the powers 2, 8, 64, 
we get 284, 18416, 9437056, whose aliquot parts make 220, etc. Thus the 
third pair is 
9363584 = 2^-191-383, 9437056 = 2"-73727. 
Descartes^^^ stated that Fermat's rule agrees exactly with his own. 
Although we saw that Mersenne quoted in 1637 the rule in Fermat's 
form and expressly attributed it to Fermat, curiously enough Mersenne^ ^^ 
gave in 1639 the rule in Descartes' form, attributing it to "un excellent 
Geometre" (meaning without doubt Descartes, according to C. Henry^"), 
''^Manuscript in Biblioth^que Nationale Paris, a commentary on the arithmetic Talkhys of 
Ibn Albanna (13th cent.). Cf. E. Lucas, L'arithm^tique amusante, Paris, 1895, p. 64. 
'"Quesiti et Inventione, 1554, fol. 98 v. 
'^♦Oeuvres de Fermat, 2, 1894, p. 72, letter to Roberval, Sept. 22, 1636; p. 208, letter to Frenicle, 
Oct. 18, 1640. 
»^euvre8 de Descartes, 2, 1898, 93-94, letter to Mersenne, Mar. 31, 1638. 
•Evidently the numbers (1) if the initial power of 2 be 2""^ 
"•Oeuvres de Descartes, 2, 1898, 148, letter to Mersenne, May 27, 1638. 
"'Bull. Bibl. Storia So. Mat. e Fis., 12, 1879, 523. 
