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Old 2008-02-04, 22:12   #1
davieddy
 
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Default 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
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Old 2008-02-04, 22:24   #2
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Yes and no. ;)

What do you mean by "as likely"?

Last fiddled with by Zeta-Flux on 2008-02-04 at 22:24
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Old 2008-02-04, 22:27   #3
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Quote:
Originally Posted by Zeta-Flux View Post
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
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Old 2008-02-04, 22:33   #4
fivemack
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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.

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Old 2008-02-04, 22:41   #5
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Quote:
Originally Posted by fivemack View Post
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
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Old 2008-02-04, 23:32   #6
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Quote:
Originally Posted by fivemack View Post
(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.
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Old 2008-02-05, 02:29   #7
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Quote:
Originally Posted by davieddy View Post
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.
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Old 2008-02-05, 17:56   #8
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Quote:
Originally Posted by davieddy View Post
"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

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Old 2008-02-05, 19:18   #9
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Quote:
Originally Posted by davieddy View Post
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
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Old 2008-02-06, 13:38   #10
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Quote:
Originally Posted by Zeta-Flux View Post
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
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Old 2008-02-06, 23:31   #11
ewmayer
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http://mathworld.wolfram.com/ChebyshevBias.html
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