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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.
Does k have to be odd?
The page for Sierpinski numbers uses both k and (2k - 1). But the page on Riesel numbers seems to say k needs to be odd.
What's so neat about Sierpinski numbers?
Is there a real-life use for numbers that are excessively composite?
And, finally....
What's a Sierpinski number of the first kind?
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
... how many prime numbers left and we discover the secrets of life?
The IT section color scheme sucks.
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
I've always thought it is unfair that only odd numbers can be prime. Why not define a number to be prime if its only possible divisors are pm 1, pm itelf, OR possibly pm 2. That way we would have a much richer collection of numbers to consider as prime.
Not trying to be sarcastic, but I have seen tons of "math theorems" and I guess I am not geeky enough to understand the point.
There's no need to have a point because the assumption is that someone will eventually find a use for it. In mathematics, physics, and all sorts of other disciplines, you don't look at your discovery and say "This would be great for X!" You publish it, forget about it, and then someone else years later has a problem to solve and does a literature search.
In the mid-1800s some poor sod went and developed a whole branch of mathematics called tensor calculus. It was an absolute mess and no one used it for anything.
Until fifty years later, Einstein is having trouble formalizing relativity and talks to his mathematician friend, who replies, "Oh, you know, I think I heard of something once..."
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Dum de dum.
Freedom is not the license to do what we like, it is the power to do what we ought.
Maybe not immediately.
However read some of the above stuff & links about the type of number.
Possible practical application:
In the fields of cyrptography/encryption - it is not beyond the realms of imagination to want to have a number which is known to be factorable, not necessarily having the factors, but very large. 78557*2^<huge number> + 1 would then be very handy. There is also a search on somewhere for more of these numbers.
Less obvious:
Symmetries, algebraic topology stuff. While I know almost completely nothing (I don't even know enough to brag about my ignorance), as people dig deeper into the links between calculus and number theory - eg Andrew Wiles and Fermat's Last Theorem proof is based on his proof of the Taniyama's Conjecture, what is slowing evolving is some understanding that will hopefully become more accessible of the links between number theory and calculus. What type of symmetries can 78557*2^<>+1 have and does it tell us anything about the algebraic topology stuff related to its construction which may then feed down into calculus (& even possibly physics).
What has me interested is why is 78557 the smallest number with this property? Why is it so special? Hopefully there may be answer before I die.
The Singularity is closer than you think
Quant