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SB Project Announces 4th-Largest Known Prime
alien88 writes "The Seventeen or Bust project announced today that they have discovered the fourth largest prime on record. The prime is 1,521,561 digits long and is their sixth discovery since the start of the project. They now have 11 multipliers left to prove that k = 78,557 is the smallest Sierpinski number. Randy Sundquist of Team ExtremeDC's computer discovered the number on December 6th."
In a related story, the BCS rankings for prime numbers were also released, with "2" garnering the top spot. Consequently, a lot of journalists got pissed off.
Chris Caldwell's page pretty much answers your questions concerning Sierpinski numbers and Risel numbers.
e rp inskiNumber
e se lNumber
http://primes.utm.edu/glossary/page.php?sort=Si
http://primes.utm.edu/glossary/page.php?sort=Ri
It is believed that 78557 is the smallest Sierpinski number, and that is what we are trying to prove. There were 17 values, when this project started, that a prime had not been found in. We are working on finding a prime in these values (11 remaining) which will then prove that 78557 is, indeed, the smallest Sierpinski number. See Chris Caldwell's page for more information.
You can construct an infinite number of provable sierpinski numbers through finding what are called "covering sets". These are sets of factors that repeat in the sequence k*2^n+1, with fixed k, and variable n.
e.g. as long as k is not divisible by 3, then half of the values k*2^n+1 will be divisible by 3. For some k it will be the even n's, for other k it will be the odd n's. Either way, you've already covered half the possibilities with a known factor. Fill in 1/4 of the values by ensuring that 5 divides half of the ones not divisible by 3, hey presto - only 1/4 now remain. 17 can remove 1/8, leaving 1/8. 65537 can remove 1/16, leaving 1/16. Between them, 241, 97 and 673 can remove 1/16 (as they can each remove 1/48). That's it - there's your covering set {3,5,17,65537,241,97,673}.
Finding which k values actually use this covering set is an exercise in using the Chinese Remainder Theorem.
(note - may be errors in the above, I did it off the top of my head, but looks right.)
If you can't find a covering set, and for the remaining 11 numbers that looks most likely, then you're right, you can't know for sure that there is no prime.
YAW.
Your head of state is a corrupt weasel, I hope you're happy.
Does anyone else think www.seventeenorbust.com sounds like a porn site?
- Chuq
Wow! I'm surprised ... coming on the heels of GIMPS 6+ million digit prime. At 1.5+ million digits, it's not only the world's 4th largest known prime, but is the FIRST known prime with more than 1 million digits that's not a Mersenne prime (not of the form 2^p-1)! This is important because the primality of this form, k*2^n+1, (while still allowing some optimizations) is much harder to check than the Mersennes and their close cousins, the Generalized Fermats, who together occupy the other 7 positions in the top 8 largest known primes!
Greg
And, for those curious, the largest prime curently known is the 40th Mersenne Prime 2 to the 20,996,011 -1, which is 6,320,430 decimal digits in length. If you're wondering what that looks like, and don't mind downloading 6.3 MB, wonder no more.
Request: ECM unit, 1000 km fullerene cable, 1 tactical nuclear weapon. Reason: Birthday party for foreign dignitary.
"The Sierpinski Problem: Definition and Status In 1960 Waclaw Sierpinski (1882-1969) proved the following interesting result.
Theorem [S]. There exist infinitely many odd integers k such that k*2n + 1 is composite for every n > 1.
A multiplier k with this property is called a Sierpinski number. The Sierpinski problem consists in determining the smallest Sierpinski number. In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is now believed to be in fact the smallest such number.
Conjecture. The integer k = 78557 is the smallest Sierpinski number.
To prove the conjecture, it suffices to exhibit a prime k*2n + 1 for each k less than 78557. By August 1997, this had been done for all except the following 21 values of k less than 78557. As long as a prime is not found for a listed k, that k might be considered a potential Sierpinski number. However, as the conjecture suggests, in the long run a prime is expected to emerge for each of these k."
So, what these folks have done is found a prime for another candidate k less than 78557.
I find the search for primes -- and for more complicated results, like this one, that use primes -- to be fascinating. There is something so pure about this world of mathematics. (As Kronecker is quoted as saying, "God made the integers; all else is the work of Man.") This kind of study says something very deep about the nature of the universe we live in.
If there are other intelligent beings in the universe, it is fascinating to contemplate that -- no matter what other differences we may have -- they may be finding out these same facts about pure mathematics. It's a language we have in common.
The Law of Falling Bodies