In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a sublime number is a positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
which has a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
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