AMICABLE BENCHES

AMICABLE BENCHES, in Roman Antiquity, were, according to Pitiscus, lower and less honourable seats allotted for the judices pedanei, or inferior judges, who, upon being admitted to the emperor's council, were dignified by him with the title amici.

AMICABLE Numbers denote pairs of numbers, of which each is mutually equal to the sum of all the aliquot parts of the other. So the first or least pair of amicable numbers

are 220 and 284; all the aliquot parts of which with their sums, are as follows, viz.:

The second pair of amicable numbers are 17,296 and 18,416, which have also the same property as above.

And the third pair of amicable numbers are 9,363,584 and 9,437,056.

These three pairs of amicable numbers were found out by F. Schooten (sect. 9 of his Exercitationes Mathematicae), who, it is said, first gave the name of amicable to such numbers, though such properties of numbers, it seems, had before been treated of by Rudolphus, Descartes, and others.

To find the first pair, Schooten puts $4x$ and $4yz$, or $a^2x$ and $a^2yz$ for the two numbers, where $a=2$; then making each of these equal to the sum of the aliquot parts of the other, gives two equations, from which are found the values of $x$ and $z$, and consequently assuming a proper value for $y$, the two amicable numbers themselves $4x$ and $4yz$.

In like manner for the other pairs of such numbers; in which he finds it necessary to assume $16x$ and $16yz$ or $a^4x$ and $a^4yz$ for the second pair, and $128x$ and $128yz$, or $a^6x$ and $a^6yz$ for the third pair.

Schooten then gives this practical rule, from Descartes, for finding amicable numbers, viz., assume the number 2, or some power of the number 2, such that if unity or 1 be subtracted from each of these three following quantities, viz.—

From 3 times the assumed number,

Also from 6 times the assumed number,

And from 18 times the square of the assumed number, the three remainders may be all prime numbers; then the last prime number being multiplied by double the assumed number, the product will be one of the amicable numbers sought, and the sum of its aliquot parts will be the other. That is, if $a$ be put = the number 2, and $n$ some integer number, such that $3a^n\; -\; 1$, and $6a^n\; -\; 1$, and $18a^\{2n\}\; -\; 1$, be all three prime numbers; then is $18a^\{2n\}\; -\; 1\; \backslash times\; 2a^n$ one of the amicable numbers, and the sum of its aliquot parts is the other. On this subject see Euleri Opuscula varii Argumenti, tom. ii. p. 23-107.