Q: WTF are
these "Sierpinski numbers"???
A: Sierpinski
numbers are numbers k, which, when in the form: (k•2^n)+1
are never primes/always composites.
Q: ...What?!
Okay, let's
start from the beginning:
Q: What is a
prime?
A: A prime is a number (larger than 0),
which is only divisible by 1 and itself. Examples are: 2, 3, 5, 7, 11, 13, 17, 19,
23....
Q: What is a
composite?
A: All numbers (above 0) that are not
primes are composites since they are divisible by another number (which then is
a factor of the composite – for example are 2, 4 and 8 all factors of 16 since
16 divided by 2, 4 or 8 produces integers)
Okay, now
back to the problem: k•2^n+1
k is an odd
integer above 0 (like 1, 3, 5, 9, 27)
n is an
integer above 0 (1, 2, 3, whatever)
k is static. n can be anything. As n gradually gets larger, sooner or
later, the outcome will be a prime. If it doesn't.... then k is a Sierpinski number.
An example:
Let's say
that k = 3
That means
the form is: (3•2^n)+1
We start at
n = 1
(3•2^1)+1 = 7 (D'oh! It's a prime!)
That means 3
is not a Sierpinski number.
The above is
a very simple example. Larger k's require more
computing.
Q: So what's
the "problem" then?
A: The Sierpinski
problem is: "What is the lowest Sierpinski
number?"
Q: Any ideas?
A: In 1962 a man named John Selfridge proposed
that the lowest Sierpinski number is 78557. But the are 39276 odd integers between 0 and 78557 and they are
all possible candidates. To prove that 78557 really is the lowest Sierpinski number requires elimination of all the 39276
candidates. That is not easy.
Q: How do I
eliminate a candidate k?
A: Pay attention! Put it in (k•2^n)+1, run through a whole bunch of n's
and if you find a prime then it is not a Sierpinski
number and it is "eliminated" as a candidate.
Q: What has
been done until now?
A: A lot of testing has been done, a lot
of primes have been found and a lot of k's have been
eliminated. In fact 39267 of the candidates have been eliminated, leaving only
8 left.
One of the
most recent elimination was of k = 27653. After running through a lot of n’s, n = 9167433 gave the form: 27653 * 2^9167433 + 1, the
result of which was a prime (since the result of this computation has 2,759,677
digits, I’m not going to write it here). That happened on June 8th
2005. A couple of primes have been found since then.
Q: Who's left?
A: 10223, 19249, 21181, 22699, 24737,
33661, 55459 and 67607
Q: How can I
help?
A: Since testing all of it yourself would
take 100 years a project has been established to try to solve the problem by
splitting it up into little pieces to each computer. The
so-called "distributed computing".
Q: What
project?
A: SoB
(Seventeen or Bust). More about it here.
Q: Why is it
taking so long? Can't I do something to speed up the process?
A: Slow computers (and fast too, if they
want) can sieve.
Q: Sieve?
A: Sieving was something Eratosthenes, an
ancient greek, invented
around 240 BC. More on him and the math in my links-section.
Q: What does it
do?
A: It runs through a whole bunch of very
large numbers to see if they divide any of the k/n combinations still left (the
Sob.dat). This process is much faster than it would
be to test them in the (k•2^n)+1. This also makes the
process of finding primes in (k•2^n)+1 easier because
of the many k/n combinations that have been eliminated. However, sieving will not
find any primes, just save the regular tests a lot of extra work.
Q: Is there any
other way of finding factors?
A: Yes, there is another process called P-1
(P minus one) factoring, but that is much more advanced, so stick with sieving
until you are no longer n00b.
Q: Okay so:
k's which doesn't ever produce primes in
(k•2^n)+1 are Sierpinski numbers?
There are
nine candidate k's left?
Sieving is
eliminating k/n pairs to make things faster?
A: Yes!