PERFECT Number
, is one that is equal to the sum of all its aliquot parts, when added together. Eucl. lib. 7, def. 22. As the number 6, which is , the sum of all its aliquot parts; also 28, for , the sum of all its aliquot parts.
It is proved by Euclid, in the last prop. of book the 9th, that if the common geometrical series of numbers 1, 2, 4, 8, 16, 32, &c, be continued to such a number of terms, as that the sum of the said series of terms shall be a prime number, then the product of this sum by the last term of the series will be a perfect number.
This same rule may be otherwife expressed thus: If n denote the number of terms in the given series 1, 2, 4, 8, &c.; then it is well known that the sum of all the terms of the series is , and it is evident that the last term is 2n - 1: consequently the rule becomes thus, viz, = a perfect number, whenever is a prime number.
Now the sums of one, two, three, four, &c, terms of the series 1, 2, 4, 8, &c, form the series 1, 3, 7, 15, 31, &c; so that the number will be found perfect whenever the corresponding term of this series is a prime, as 1, 3, 7, 31, &c. Whence the table of perfect numbers may be found and exhibited as follows; where the 1st column shews the number of terms, or the value of n; the 2d column is the last term of the series 1, 2, 4, 8, &c, and is expressed by 2n - 1; the 3d column contains the corresponding sums of the said series, or the values of the quantity ; which numbers in this 3d column are easily constructed by adding always the last number in this column to the next following number in the 2d column: and lastly, the 4th column shews the correspondent Persect Numbers, or the values of , the product of the numbers in the 2d and 3d columns, when , or the number in the 3d column, is a prime number; the products in the other cases being omitted, as not Perfect Numbers.
| Values of n | Values of 2n - 1 | Values of | Perf. Numbers, or |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 3 | 6 |
| 3 | 4 | 7 | 28 |
| 4 | 8 | 15 | . |
| 5 | 16 | 31 | 496 |
| 6 | 32 | 63 | . |
| 7 | 64 | 127 | 8128 |
Hence the first four Perfect Numbers are found to be 6, 28, 496, 8128; and thus the table might be continued to find others, but the trouble would be very great, for want of a general method to distinguish which numbers are primes, as the case requires. Several learned mathematicians have endeavoured to facilitate this business, but hitherto with only a small degree of perfection. After the foregoing four Perfect Numbers, there is a long interval before any more occur. The first eight are as follow, with the factors and products which produce them:
| The first Perfect Numbers. | Their values. |
| 6 | = (22 - 1) 2 |
| 28 | = (23 - 1) 22 |
| 496 | = (25 - 1) 24 |
| 8128 | = (27 - 1) 26 |
| 33550336 | = (213 - 1) 212 |
| 8589869056 | = (217 - 1) 216 |
| 137438691328 | = (219 - 1) 218 |
| 2305843008139952128 | = (231 - 1) 230 |
See several considerable tracts on the subject of Perfect Numbers in the Memoirs of the Petersburgh Academy, vol. 2 of the new vols, and in several other volumes.
PERIÆCI. See Perioeci.
PERIGÆUM, or Perigee, is that point of the orbit of the sun or moon, which is the nearest to the earth. In which sense it stands opposed to Apogee, which is the most distant point from the earth.|
Perigee, in the Ancient Astronomy, denotes a point in a planet's orbit, where the centre of its epicycle is at the least distance from the earth.
