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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."
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.
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.
I've always thought it is unfair that only odd numbers can be prime.
2 is both even and prime.
"They redundantly repeated themselves over and over again incessantly without end ad infinitum" -- ibid.