AMBLIGON

, or Ambligonal, in Geometry, signisies obtuse-angular, as a triangle which has one of its angles obtuse, or consisting of more than 90 degrees.

AMICABLE numbers, denote pairs of numbers, of which each of them 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 follow, viz, of 220, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, their sum — — 284; of 284, they are 1, 2, 4, 71, 142, and their sum is 220.

The 2d pair of amicable numbers are 17296 and 18416, which have also the same property as above.

And the 3d pair of amicable numbers are 9363584 and 9437056.

These three pairs of amicable numbers were found out by F. Schooten, sect. 9 of his Exercitationes Mathematicæ, who I believe 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 sirst pair, Schooten puts 4x and 4yz, or a²x and a²yz 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 a⁴x and a⁴yz for the 2d pair, and 128x and 128yz or a⁷x and a⁷yz for the 3d 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 3aⁿ-1, and 6aⁿ-1, and 18a²ⁿ-1 be all three prime numbers; then is ―(18a²ⁿ-1) X2aⁿ one of the amicable numbers; and the sum of its aliquot parts is the other.