Today's number is
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
This number is the largest known sublime number! But to understand what a sublime number is, we need to know what a perfect number is.
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 example, 6 has proper divisors 1, 2, and 3, and
1 + 2 + 3 = 6.
The next perfect number is 28, which has proper divisors 1, 2, 4, 7, and 14, and
A sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number.
12 is an example of a sublime number. It has 6 positive divisors, and 6 is a perfect number:
and the sum of these is a perfect number:
1 + 2 + 3 + 4 + 6 + 12 = 28
As of July 2026, there are only two known sublime numbers, namely 12 and 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
We can show that this number is a sublime number by the Euclid-Euler theorem. It states that a positive integer is an even perfect number if and only if it has the form 2^{p-1}(2^p - 1), where 2^p - 1 is a prime number.
In our case, that huge number has 8128 positive divisors, and
where 2^7 - 1 is prime. By the Euclid-Euler theorem, 8128 is therefore a perfect number.
Even more remarkably, these 8128 divisors add up to
which is the 12th even perfect number because 2^{127} - 1 is the 12th Mersenne prime.
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