Mersenne Digest Saturday, April 7 2001 Volume 01 : Number 837
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Date: Thu, 5 Apr 2001 06:38:38 +0200 (MET DST)
From: [EMAIL PROTECTED]
Subject: Re: Mersenne: arithmetic progression of consecutive primes
> From: "Gary Untermeyer" <[EMAIL PROTECTED]>
> Greetings,
> Although this is not a question regarding mersenne primes, I thought I'd
> throw this out to the readers here.
> Let x be a prime number. Consider the series of numbers that take the
> following form:
> x, x + n, x + 2n, x + 3n, x + 4n, x + 5n, x + 6n, where n is an
> even positive whole number.
> In this series of seven numbers, can anyone tell me why, if ALL of these
> numbers are prime, that the minimum value of n is 210 if all the terms
> are _consecutive_ prime numbers?
The _consecutive_ hypothesis is not required,
other than requiring x > 7 and n > 0. If n is not divisible by 7,
then one of the seven numbers x, x + n, ..., x + 6n
will be divisible by 7, hence not prime.
[No two of the seven will be congruent modulo 7,
so all possible remainders modulo 7 must be represented.]
Likewise n must be divisible by 2, 3, 5.
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Date: Thu, 5 Apr 2001 19:33:22 +0200
From: =?iso-8859-1?Q?Torbj=F6rn_Alm?= <[EMAIL PROTECTED]>
Subject: Re: Mersenne: arithmetic progression of consecutive primes
Greetings!
There are chains known today of at least 22 primes in arithmetic
progression.
Paul Pritchard has found a number of them.
1968 I searched for such PAP:s as they are known together with Hans Riesel,
but the available computer power at that time was too hard to get.
Later I picked up the old project, and I have a fairly large collection of
PAP:s.
Torbj�rn Alm
- ----- Original Message -----
From: <[EMAIL PROTECTED]>
To: "Gary Untermeyer" <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]>
Sent: Thursday, April 05, 2001 6:38 AM
Subject: Re: Mersenne: arithmetic progression of consecutive primes
> > From: "Gary Untermeyer" <[EMAIL PROTECTED]>
>
> > Greetings,
>
> > Although this is not a question regarding mersenne primes, I thought I'd
> > throw this out to the readers here.
>
> > Let x be a prime number. Consider the series of numbers that take the
> > following form:
>
> > x, x + n, x + 2n, x + 3n, x + 4n, x + 5n, x + 6n, where n is an
> > even positive whole number.
>
> > In this series of seven numbers, can anyone tell me why, if ALL of these
> > numbers are prime, that the minimum value of n is 210 if all the terms
> > are _consecutive_ prime numbers?
>
> The _consecutive_ hypothesis is not required,
> other than requiring x > 7 and n > 0. If n is not divisible by 7,
> then one of the seven numbers x, x + n, ..., x + 6n
> will be divisible by 7, hence not prime.
> [No two of the seven will be congruent modulo 7,
> so all possible remainders modulo 7 must be represented.]
> Likewise n must be divisible by 2, 3, 5.
>
>
>
>
>
> _________________________________________________________________________
> Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
> Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
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------------------------------
Date: Fri, 06 Apr 2001 19:32:14 +0200
From: Henk Stokhorst <[EMAIL PROTECTED]>
Subject: Re: Mersenne: graphical image of factoring work being done
L.S.,
The program I wrote to track other people's work in factoring exponents
in and outside the server assigned ranges has now been updated too. It
is now based on the same 'core' as the program used to generate a
graphical overview of all factoring work done.
If you downloaded htpp://home.wxs.nl/~tha/changes_overview.zip then
please upgrade to the new version, it is more flexible. Delphi source
code is included.
YotN,
Henk.
PS. Drop a note if you find the program useful, want a change or use a
different method to check other peoples activities in not server
assigned ranges.
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------------------------------
Date: Sat, 7 Apr 2001 13:15:16 EDT
From: [EMAIL PROTECTED]
Subject: Re: Mersenne: arithmetic progression of consecutive primes
In a message dated 05/04/2001 05:19:38 GMT Daylight Time, "Gary Untermeyer"
<[EMAIL PROTECTED]> writes:
> Greetings,
> Although this is not a question regarding mersenne primes, I thought I'd
> throw this out to the readers here.
You will find [EMAIL PROTECTED] is also good for this kind of
question.
> Let x be a prime number. Consider the series of numbers that take the
> following form:
>
> x, x + n, x + 2n, x + 3n, x + 4n, x + 5n, x + 6n, where n is an
> even positive whole number.
>
> In this series of seven numbers, can anyone tell me why, if ALL of these
> numbers are prime, that the minimum value of n is 210 if all the terms
> are _consecutive_ prime numbers?
If n is not a multiple of 2 (i.e. even), then one of the first 2 terms
x, x+n
is (obviously) a multiple of 2 and so is not prime.
Similarly, if n is not a multiple of 3, then one of first 3 terms
x, x+n, x+2n
is a multiple of 3 and so is not prime.
And so on, for primes 2, 3, 5, ...
Inverting the argument, if the sequence has N terms, then n must be a
multiple of all the primes <= N. So in your example (N = 7), n must be a
multiple of 2, 3, 5 and 7, i.e. of 210. This is still the case for N = 8
(your next case), 9, and 10. For N=11, n must be a multiple of 210*11=2310.
Note that this is true for _all_ arithmetic progressions of primes, not just
for progressions of _consecutive_ primes.
Mike Oakes
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End of Mersenne Digest V1 #837
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