## Re: [PrimeNumbers] Checking Large "Prime Numbers"?

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• ... With Pari/GP in a fraction of a second: ? test(p)=centerlift(6*Mod(10,p)^1000000000)^2 ? forprime(p=2,100000,if(test(p)==1,print(p))) 31 293 2861 12547
Message 1 of 9 , May 8, 2006
--- Bob Gilson <bobgillson@...> wrote:
> A colleague of mine claimed the other day that
>
> 5, followed by one billion 9's, and 6, followed by 999,999,999 zeroes, with
> a further last digit being 1, are in fact twin primes.
>
> How does anyone go about refuting or confirming such allegations?

With Pari/GP in a fraction of a second:
? test(p)=centerlift(6*Mod(10,p)^1000000000)^2
? forprime(p=2,100000,if(test(p)==1,print(p)))
31
293
2861
12547
time = 60 ms.

Your colleague is a rank amateur at such claims.

I suspect that there are as we speak only three people in the world
(Peter Montgomery, Dave Rusin, myself) that have ever seen a proof
of the truth or falsity of the following statement:

3^(3^20)+4 is prime.

Feel free to simply turn the tables on your colleague and recycle
my example back at him. (It's not my claim, an anonymous 4th person
made the claim but I don't believe he had any proof either way.)

A Usenet archive should contain some clues as to the real solution.

Phil

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• ... with ... Impressive timings! This is the only response that actually seemed to answer the original question, *how does one go about it* rather than just
Message 2 of 9 , May 8, 2006
Phil Carmody wrote:
> --- Bob Gilson <bobgillson@...> wrote:
> > A colleague of mine claimed the other day that
> > 5, followed by one billion 9's, and 6, followed by 999,999,999 zeroes,
with
> > a further last digit being 1, are in fact twin primes.
> > How does anyone go about refuting or confirming such allegations?
>
> With Pari/GP in a fraction of a second:
> ? test(p)=centerlift(6*Mod(10,p)^1000000000)^2
> ? forprime(p=2,100000,if(test(p)==1,print(p)))

Impressive timings! This is the only response that actually seemed to
answer the original question, *how does one go about it* rather than just
enigmatically listing factors, which does not help the original poster, nor
answer the question posed. Could you explain the mathematics behind this one
(especially why you use the centerlift function and what it does) for those
not familiar with Pari/GP?

I would also be interested if the others who posted results would answer
the original question--how one goes about testing claims like this
(efficiently, I hope. I know how to do it several brute-force ways.) Thanks!

--
Alan Eliasen
eliasen@...
http://futureboy.us/
• ... Woo-woo! Brownie-points for Phil! I was thinking of answering by evaluating the expressions for the two numbers modulo 31 , which could have been a
Message 3 of 9 , May 8, 2006
--- Alan Eliasen <eliasen@...> wrote:
> Phil Carmody wrote:
> > --- Bob Gilson <bobgillson@...> wrote:
> > > A colleague of mine claimed the other day that
> > > 5, followed by one billion 9's, and 6, followed by 999,999,999 zeroes,
> with
> > > a further last digit being 1, are in fact twin primes.
> > > How does anyone go about refuting or confirming such allegations?
> >
> > With Pari/GP in a fraction of a second:
> > ? test(p)=centerlift(6*Mod(10,p)^1000000000)^2
> > ? forprime(p=2,100000,if(test(p)==1,print(p)))
>
> Impressive timings! This is the only response that actually seemed to
> answer the original question, *how does one go about it* rather than just
> enigmatically listing factors, which does not help the original poster, nor

Woo-woo! Brownie-points for Phil!

I was thinking of answering "by evaluating the expressions for the two numbers
modulo 31", which could have been a middle-ground between my useful :-) and

> Could you explain the mathematics behind this one
> (especially why you use the centerlift function and what it does) for those
> not familiar with Pari/GP?

It simply picks a distinguished member of the set of numbers == a (mod b) in
the range (-b/2, b/2] rather than [0,b). So rather than +1 and p-1 you'll have
-1 and +1. Hence the square to subsequently turn both of those into 1.

> I would also be interested if the others who posted results would answer
> the original question--how one goes about testing claims like this
> (efficiently, I hope. I know how to do it several brute-force ways.)
> Thanks!

Essentially, the same way as the above, I'd bet.

If you've been given a large number that is claimed to be prime, and it's not
obviously a product of hand-crafted secret numbers, then the best way to
counter the claim of primality is almost always to find a small factor. The
best way to find a small factor is to evaluate the expression for it modulo the
small primes in turn.

Phil

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• Thanks to everyone for the great response - now for the joys of PARI/GP Regards Bob ... Woo-woo! Brownie-points for Phil! I was thinking of answering by
Message 4 of 9 , May 9, 2006
Thanks to everyone for the great response - now for the joys of PARI/GP

Regards

Bob

Phil Carmody <thefatphil@...> wrote:
--- Alan Eliasen <eliasen@...> wrote:
> Phil Carmody wrote:
> > --- Bob Gilson <bobgillson@...> wrote:
> > > A colleague of mine claimed the other day that
> > > 5, followed by one billion 9's, and 6, followed by 999,999,999 zeroes,
> with
> > > a further last digit being 1, are in fact twin primes.
> > > How does anyone go about refuting or confirming such allegations?
> >
> > With Pari/GP in a fraction of a second:
> > ? test(p)=centerlift(6*Mod(10,p)^1000000000)^2
> > ? forprime(p=2,100000,if(test(p)==1,print(p)))
>
> Impressive timings! This is the only response that actually seemed to
> answer the original question, *how does one go about it* rather than just
> enigmatically listing factors, which does not help the original poster, nor

Woo-woo! Brownie-points for Phil!

I was thinking of answering "by evaluating the expressions for the two numbers
modulo 31", which could have been a middle-ground between my useful :-) and

> Could you explain the mathematics behind this one
> (especially why you use the centerlift function and what it does) for those
> not familiar with Pari/GP?

It simply picks a distinguished member of the set of numbers == a (mod b) in
the range (-b/2, b/2] rather than [0,b). So rather than +1 and p-1 you'll have
-1 and +1. Hence the square to subsequently turn both of those into 1.

> I would also be interested if the others who posted results would answer
> the original question--how one goes about testing claims like this
> (efficiently, I hope. I know how to do it several brute-force ways.)
> Thanks!

Essentially, the same way as the above, I'd bet.

If you've been given a large number that is claimed to be prime, and it's not
obviously a product of hand-crafted secret numbers, then the best way to
counter the claim of primality is almost always to find a small factor. The
best way to find a small factor is to evaluate the expression for it modulo the
small primes in turn.

Phil

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• May I add a little more explanation to Phil s Pari script for us who are still novices? We re trying to find factors of n +/- 1 where n=6*10^1000000000.
Message 5 of 9 , May 10, 2006
May I add a little more explanation to Phil's Pari script for us who are
still novices?
We're trying to find factors of n +/- 1 where n=6*10^1000000000. Phil's
script calculates
n (mod p) for primes p from 2 to 100000 and then squares it. If the
result is 1, then n (mod p)
is either -1 or +1. If it is -1, p is a factor of n+1 and if n(mod p) is
+1, then p is a
factor of n-1. Squaring combined these two tests into one.

Here was Phil's script:

test(p)=centerlift(6*Mod(10,p)^1000000000)^2
forprime(p=2,100000,if(test(p)==1,print(p)))

For Pari novices, like myself, this is a good example of how to use IntMod
types, which
is what you get with the Mod(x,p) function. When you do arithmetic
functions on an IntMod, the
result is always calculated modulo p, so it never gets too big. Now,
10^1000000000 is too big
for Pari to handle, but Mod(10,p)^100000000 does not get too big. Pari
will do this
exponentiation without overflowing anything. Same with the multiply by 6.

An IntMod type is always in the range of 0 to p-1 (mod p), and lift()
converts that type to
an integer in that range. But centerlift( ) converts it to an integer in
the
range (-b/2, b/2], as Phil explained. p-1 is now -1, p-2 would be -2,
etc.

Phil could have made his test use lift() and then compare test(p) to 1 OR
p-1 which would
require a temp variable but eliminate needing to square. Only Phil would
know which
would be faster. Here's how that could be implemented.

test(p)=lift(6*Mod(10,p)^1000000000)
forprime(p=2,100000,temp=test(p);\
if(temp==1,print(p," is a factor of n-1"));\
if(temp==(p-1),print(p," is a factor of n+1")))

I had to put \ at the end of some lines -- I still don't know the rules
they are necessary.

Hope this helps.

primenumbers@yahoogroups.com wrote on 05/09/2006 10:01:04 AM:

> Thanks to everyone for the great response - now for the joys of PARI/GP
>
> Regards
>
> Bob
>
> Phil Carmody <thefatphil@...> wrote:
> --- Alan Eliasen <eliasen@...> wrote:
> > Phil Carmody wrote:
> > > --- Bob Gilson <bobgillson@...> wrote:
> > > > A colleague of mine claimed the other day that
> > > > 5, followed by one billion 9's, and 6, followed by 999,999,999
zeroes,
> > with
> > > > a further last digit being 1, are in fact twin primes.
> > > > How does anyone go about refuting or confirming such
allegations?
> > >
> > > With Pari/GP in a fraction of a second:
> > > ? test(p)=centerlift(6*Mod(10,p)^1000000000)^2
> > > ? forprime(p=2,100000,if(test(p)==1,print(p)))
> >
> > Impressive timings! This is the only response that actually seemed
to
> > answer the original question, *how does one go about it* rather than
just
> > enigmatically listing factors, which does not help the original
poster, nor
> > answer the question posed.
>
> Woo-woo! Brownie-points for Phil!
>
> I was thinking of answering "by evaluating the expressions for the two
numbers
> modulo 31", which could have been a middle-ground between my useful :-)
and
> everyone else's useless :-P answers.
>
> > Could you explain the mathematics behind this one
> > (especially why you use the centerlift function and what it does) for
those
> > not familiar with Pari/GP?
>
> It simply picks a distinguished member of the set of numbers == a (mod
b) in
> the range (-b/2, b/2] rather than [0,b). So rather than +1 and p-1
you'll have
> -1 and +1. Hence the square to subsequently turn both of those into 1.
>
> > I would also be interested if the others who posted results would
> > the original question--how one goes about testing claims like this
> > (efficiently, I hope. I know how to do it several brute-force ways.)
> > Thanks!
>
> Essentially, the same way as the above, I'd bet.
>
> If you've been given a large number that is claimed to be prime, and
it's not
> obviously a product of hand-crafted secret numbers, then the best way to
> counter the claim of primality is almost always to find a small factor.
The
> best way to find a small factor is to evaluate the expression for
itmodulo the
> small primes in turn.
>
> Phil
>

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