New Record Prime Found

An anonymous reader writes, "The GIMPS project has found a new record prime. 2 ^ 32,582,657 - 1, clocking in at over 9 million digits, is a Mersenne prime, as were the last few record primes. Here is the 9-megabyte decimal expansion."

Re:Help

[publius1234]

Mersenne primes are simply primes that have this form, but not all numbers that have this form are prime. For example, 15 = 2^4 -1 and is not prime.

Also, for a number to be a Mersenne prime, the 'n' from 2^n-1 must be prime (according to Mathworld [wolfram.com]).

Expand it yourself

[Coppit]

If you want to expand the number on your machine, run this:

perl -Mbignum -e 'print 2**32582657 - 1'

If it takes too long for you, you can also have perl print an approximation:

perl -e 'print 2**32582657 - 1'

Re:Help

[fatphil]

If a divides n, then 2^a-1 divides 2^n-1.
So for 2^n-1 to be prime, n itself must be prime.

Quick proof - consider the values of 2^i modulo (2^a-1) for i=0..n,
You'll notice that 2^0 == 2^a == 2^(2a) == .... 2^((n/a)*a) == 1.
i.e. 2^n-1 == 0 (mod 2^a-1)

Note, however, that it's a necessary condition, but is not sufficient.
There are plenty of prime p such that 2^p-1 is not prime.

See http://www.primepages.org/ [primepages.org]

FatPhil

Re:You know what would be funny?

[kestasjk]

I know that was a joke, but this gives the impression that this number in binary is a long stream of 1's and 0's.

Seeing as 2^n is [1 followed by n 0's] in binary, and [1 followed by n 0's] minus 1 is [(n-1) 1's], this number in binary will just be 32,582,656 1's, which isn't decodeable as an MP3.

The Actual Number

[neonprimetime]

View it here [isthe.com] in it's 12mb text form!!!!!!!!

Re:It's nice to see, but...

[Ruzty]

Folding@Home is still pretty neat though - the whole "use your spare CPU cycles to (potentially) find cures for various nasty things"

I think you should have said:
Folding@Home is still pretty neat though - the whole "use your spare CPU cycles to (potentially) find cures for various nasty things and enrich the patent portfolios of drug manufacturers"

HTH HAND
-R

Re:A good definition

[jd]

Any natural number that is a prime can be split into a sum of natural numbers (where the number of elements in the sum is at least 2) for all natural numbers that are primes other than 1. It is always possible to have one of those numbers non-prime (except for the natural numbers 2 and 3). By dividing the non-prime numbers into the prime factors, and then repeating the process, you can split the original number into some expression e, where if we swap the constants for variables (or unknowns) the general form is E. There will be a number of possible values of E, so I'll call the set of all possible expressions for that prime {E}. Ok, now let us say that the set of all possible sets of expressions for all possible primes is {{E}*}. Because I'm allowing any general solution, there will be a finite number of generalized solutions (ie: using undefined constants) but potentially an infinite number of specialized solutions where the constants are all defined.

5, for example, is a solution to the general formula A(x^C)+D, sum(x)+C, Cx+D, and so on. (Because A, C and D are constants, they would need to be the same constants for all primes you apply the expression to. Otherwise, you've not really generalized or simplified anything.) It is easy to show this has an infinite number of special solutions - pick any value of C and for any equation with D in it, simply subtract/add the necessary value to give you the prime you want.

The original question, then, is whether there is any prime number (other than the three special cases listed) for which {E} /\ {{E}*} = {E}. That would be a prime for which no possible expression to derive it, no matter how generalized, can ever be used to derive any other prime number. Intuitively, it seems apparent that no such prime exists, as there will be many primes with an infinite number of solutions that do not overlap. However, there are also an infinite number of primes and I have no obvious way of proving that the infinite set of primes is a perfect subset of the infinite number of derivations.

Let us define something else, which is #({E} /\ {{E}*}) - the number of general solutions for a given prime that are also general solutions for at least one other prime. The larger the prime, the larger the number of possible generalized solutions, because the larger the number of subcomponents you can break it into which can be turned into equations. However, the number of overlaps is not so easily determined. It is very easy to imagine a large prime in which most general forms are the same as only a few other primes, whereas a smaller prime might overlap with a great many primes in a great many ways. It is also possible to imagine a set of primes in which all primes in that set have the same general form, but no member has a general form that is the same as any outside of that set.

(It is easy to prove that there exist two primes which have no general form in common, as that is simply the same as the proof that no general equation for primes exists.)

This leads to interesting possibilities - "islands" of primes that are totally disconnected from all other primes, "peninsulas" where you almost have an island but some perfect subset of the cloud does have a general form in common with other primes, "mountains" where you have a massive number of general forms totally in common with a large number of primes, and so on.

If you were to draw out the interrelationships between primes as a topology, what do you see? A random blob? A sea of islands? Multiple large masses that are otherwise disconnected? If islands, or multiple disconnected continents, exist in prime number space, does this mean that prime numbers aren't a single, definable set of numbers at all, but multiple concepts that should (in general) not be treated as the same at all? Will the map indicate that we can generalize the definition of prime number in a useful way, that the concept can be usefully extended and meaningfully applied outside of the natural numbers?