Classical Arithmetic, Lesson 15. Book I, Chapter 15

The following tasks are required to complete this lesson:

1. 1. Study the Lesson carefully.
   2. Complete the lesson Assessment.

Lesson

On account of the paucity, therefore, of perfect numbers, there is only one between 1 and 10, viz. 6; one only between 10 and 100, viz. 28; between 100 and 1,000 only one, 496; and between 1,000 and 10,000 the only perfect number is 8,128.  These numbers likewise, are always terminated by the two even numbers 6 and 8, as is evident in those already adduced.

But the generation of them is fixed and firm, and can only be effected in one way. For evenly-even numbers being disposed in an orderly series from unity, the first must be added to the second, and if a first and incomposite number is produced by that addition, this number must be multiplied by the second of the evenly-even numbers, and the product will be a perfect number. If, however, a first and incomposite number is not produced by the addition, but a composite and second number, this must be passed by, and the number which follows must be added. And if this aggregate is not found to be a first and incomposite number, another must be added, and this must be done till a first number is found. When therefore this is found, it must be multiplied into the last of the added evenly-even numbers, and the product will be a perfect number.

Thus for instance, in the evenly-even series of numbers 1, 2, 4, 8, 16, 52, 64, 128, if 1 is added to 2 the sum is 3, and because 3 is a first and incomposite number, this multiplied by 2 will produce the perfect number 6.

But 28 the next perfect number is produced as follows: from the addition of 1, 2 and 4 arises 7, which is a first and incomposite number, and this being multiplied by 4 the last of the evenly-even numbers, the perfect number 28 is produced.

Since therefore these two perfect numbers 6 and 28 are found, others must be investigated after the same manner. Thus, in order to find the next perfect
number, add 1, 2, 4, and 8 together; but the sum of these is 15. This however is a second and composite number; for it has a third and a fifth part, besides a fifteenth part, unity, which is denominated from itself. This therefore must be passed by, and the next evenly-even number, viz. 16, must be added to it, and the sum will be 31, which is a first and incomposite number. Let this then be multiplied by 16 the last of the evenly-even numbers, and the product will be 496 the next perfect number after 28.

The monad therefore is in power though not in energy itself a perfect number. For if it is first assumed in the order of numbers, it will be found to be primary and incomposite, and if multiplied by itself, the same unity is produced. But this is equal to its parts in power alone. Hence the monad is perfect by its own proper virtue, is first and incomposite, and preserves itself unchanged when multiplied by itself.

The way however in which perfect numbers are generated, will immediately become manifest by the following table:

Notes:
The odd numbers in the table above are produced by the addition of the evenly-even numbers.
The asterisks in this table signify that the odd numbers above them produced by the addition of the evenly-even numbers, are composite, and therefore unfit to form perfect numbers.

Assessment

• Comprehension Questions