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 2008-02-04, 22:12 #1 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts Primes mod 4 Are they as likely to ==1 as ==3? PS thought I'd ask here rather than the proper maths thread to avoid embarrassment
 2008-02-04, 22:24 #2 Zeta-Flux     May 2003 60B16 Posts Yes and no. ;) What do you mean by "as likely"? Last fiddled with by Zeta-Flux on 2008-02-04 at 22:24
2008-02-04, 22:27   #3
davieddy

"Lucan"
Dec 2006
England

2·3·13·83 Posts

Quote:
 Originally Posted by Zeta-Flux Yes and no. ;) What do you mean by "as likely"?
Are there the ~same number of each below 100M?

Last fiddled with by davieddy on 2008-02-04 at 22:28

 2008-02-04, 22:33 #4 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 11000111010002 Posts http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf is likely to tell you all you want to know about this, and then a little more besides. It's a well-known nice problem. http://www.math.umn.edu/~focm/c_/Martin.pdf is a slightly more sophisticated article for people with a small amount of analytic number theory background (that is, who know a Dirichlet L-function from a hole in the ground), amongst whom I once counted myself but now don't. Last fiddled with by fivemack on 2008-02-04 at 22:36
2008-02-04, 22:41   #5
davieddy

"Lucan"
Dec 2006
England

145128 Posts

Quote:
 Originally Posted by fivemack http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf is likely to tell you all you want to know about this, and then a little more besides. It's a well-known nice problem. http://www.math.umn.edu/~focm/c_/Martin.pdf is a slightly more sophisticated article for people with a small amount of analytic number theory background (that is, who know a Dirichlet L-function from a hole in the ground), amongst whom I once counted myself but now don't.
THX

I take it the answer wasn't "yes" then

OTOH If I interpret "nice" as meaning "subtle", then I think
that "yes" is good enough for my present purposes.

Last fiddled with by davieddy on 2008-02-04 at 22:56

2008-02-04, 23:32   #6
davieddy

"Lucan"
Dec 2006
England

145128 Posts

Quote:
 Originally Posted by fivemack (that is, who know a Dirichlet L-function from a hole in the ground), amongst whom I once counted myself but now don't.
I'm old enough to remember Bernard Cribbins' "Hole in the ground"
but I'm sure Ernst would find something wittier to say.

2008-02-05, 02:29   #7
Zeta-Flux

May 2003

7·13·17 Posts

Quote:
 Originally Posted by davieddy THX I take it the answer wasn't "yes" then
Yes, unless you are using a logarithmic measure of how often one count is ahead of another (as defined on page 18 of the first link). In that sense 3 mod 4 beats 1 mod 4 soundly.

2008-02-05, 17:56   #8
davieddy

"Lucan"
Dec 2006
England

11001010010102 Posts

Quote:
 Originally Posted by davieddy "yes" is good enough for my present purposes.
My "purpose" was to investigate the Wagstaff conjecture.
The probability of 2^p-1 being prime involves ln(ap) where
a=2 if p==3 mod 4 and a=6 if p==1 mod 4.

Commonly we take ln(ap)~ln(p) , true for huge p.
But for p in the GIMPS range I calculate ln(ap)~1.07 ln(p),
an appreciable discrepancy.

http://primes.utm.edu/mersenne/heuristic.html

Last fiddled with by davieddy on 2008-02-05 at 18:52

2008-02-05, 19:18   #9
davieddy

"Lucan"
Dec 2006
England

647410 Posts

Quote:
 Originally Posted by davieddy a=2 if p==3 mod 4 and a=6 if p==1 mod 4. But for p in the GIMPS range I calculate ln(ap)~1.07 ln(p),
ln(65M)~18
ln(2)~0.7
ln(6)~1.8

2.5 is 7% of 36.

Last fiddled with by davieddy on 2008-02-05 at 19:26

2008-02-06, 13:38   #10
davieddy

"Lucan"
Dec 2006
England

145128 Posts

Quote:
 Originally Posted by Zeta-Flux Yes, unless you are using a logarithmic measure of how often one count is ahead of another (as defined on page 18 of the first link). In that sense 3 mod 4 beats 1 mod 4 soundly.
I think page 2 "The prime number theorem for arithmetic progressions"
told me what I needed to know.

David

 2008-02-06, 23:31 #11 ewmayer ∂2ω=0     Sep 2002 República de California 2D3216 Posts

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