In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, a sublime number is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
which has a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...
of positive
factors (including itself), and whose positive factors add up to another perfect number.
The number
12, for example, is a sublime number. It has a perfect number of positive factors (
6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 =
28.
There are only two known sublime numbers: 12 and (2
126)(2
61 − 1)(2
31 − 1)(2
19 − 1)(2
7 − 1)(2
5 − 1)(2
3 − 1) .
[ Clifford A. Pickover, ''Wonders of Numbers, Adventures in Mathematics, Mind and Meaning'' New York: Oxford University Press (2003): 215] The second of these has 76 decimal digits:
:6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.
References
{{Classes of natural numbers
Divisor function
Integer sequences