New perfect number discovered

A positive integer N is described as a perfect number if the sum of all of its proper divisors (that is to say, factors of N other than N) is equal to itself. For instance, the proper divisors of 28 are 1, 2, 4, 7 and 14, which sum to 28.

The first few perfect numbers are {6, 28, 496, 8128, 33550336, …}. Since 2008, the largest known perfect number was 2^{86225217} - 2^{43112608}. This record has now been broken, with the discovery of the perfect number 2^{115770321} - 2^{57885160}.

It is unknown as to whether there are any odd perfect numbers. An even number is perfect if and only if it is of the form (2^p - 1)2^{p-1}, where 2^p - 1 is a (Mersenne) prime. Mersenne primes are much easier to verify than ordinary primes of similar size; the Mersenne prime 2^p- 1 can be verified in time O(p^2 log(p) log(log(p))) by using the Lucas-Lehmer test together with the Schönhage-Strassen algorithm for multiplying large integers. By comparison, a non-Mersenne prime of a similar size takes time O(p^6) using the best known algorithm (AKS) for primality testing.

As of today, there are now 48 known perfect numbers. We don’t know whether there are any more, although it has been conjectured that there are infinitely many.

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One Response to New perfect number discovered

  1. my name says:

    Best post I read about the subject in the last week, since the discovery! All the other who did coverage for the finding of the new mersenne concentrated on that, writing a lot of incorrect math around it. Concentrating on the perfect number instead, which is bigger, and sounds nicer, this was a good move. However, you should talk a bit more about how the number was find, say something about Cooper, GIMPS or mersenneforum.com. 😛

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