## Consecutive primes on a prime leash

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• Consider: the prime sequence 29, 31, 37, 41, 43, 47, 53. can be regarded as 24 plus: 5, 7, 13, 17, 19, 23, 29. (The prime eleven is missing but what matters is
Message 1 of 11 , Mar 12, 2006
Consider: the prime sequence 29, 31, 37, 41, 43, 47, 53.
can be regarded as 24 plus: 5, 7, 13, 17, 19, 23, 29.

(The prime eleven is missing but what matters is that they are all
prime.)

So we are looking for basecamp numbers (like 24), after which there
is a long sequence of primes, each prime a prime distance from the
base.

How does the sequence grow? Below is a list showing (basecamp number,
length of sequence generated) for increasing sequences. Notice how
jumpy the sequence length can get:

(8,2)
(24,7) (as seen above)
(90,9)
(120,14)
(840,19)
(4410,21)
(6930,24)
(10920,26)
(20370,27)
(30030,29)
(115500,31)
(159390,40)

Just look at that jump from 31 to 40!

There are 40 consecutive primes after 159390 which can be written as
159390 plus :

(13, 17, 31, 41, 47, 67, 73, 79, 83, 101, 109, 113, 131, 149, 151,
163, 173, 179, 181, 199, 227, 233, 239, 241, 277, 281, 283, 293, 307,
311, 317, 331, 347, 349, 373, 379, 383, 389, 397, 401.)

Unsurprisingly 159390 is factor rich: 159390 = 2 * 3^2 * 5 * 7 * 11
* 23.

(Another option would be to account also for primes on the other side
of the basecamp which might increase the sequence length, sometimes
dramatically. For instance 120 has 14 primes above and 10 primes
below it, making a sequence of 24 consecutive primes, each a prime
away from 120.)

Longer sequences, anybody?

Mark
• Talk about another jump. After (159390,40) the next increases are (2106720,43) and (3573570,53)! So 3573570 has 53 primes immediately after it, such that each
Message 2 of 11 , Mar 13, 2006
Talk about another jump. After (159390,40) the next increases are
(2106720,43) and (3573570,53)!

So 3573570 has 53 primes immediately after it, such that each prime
is a prime from 3573570. The primes are:

3573570 + (29, 37, 43, 53, 59, 67, 103, 109, 127, 139, 157, 163,
179, 181, 191, 199, 229, 233, 251, 257, 269, 307, 317, 347, 367, 383,
401, 409, 443, 467, 479, 491, 509, 521, 541, 557, 571, 587, 599, 617,
619, 631, 661, 677, 683, 727, 739, 743, 773, 787, 809, 811, 821).

It so happens that 3573570 also has 26 primes immediately before it
such that each prime is a prime away from 3573570. The primes are:

3573570 - (41, 43, 47, 53, 59, 61, 67, 71, 79, 97, 101, 109, 139,
157, 167, 179, 197, 229, 233, 239, 307, 311, 317, 331, 347, 349).

That would make 79 consecutive primes, each a prime away from 3573570.

As would be expected, 3573570 is very composite!
3573570 = 2 * 3 * 5 * 7^2 * 11 * 13 * 17

Mark
• I can t believe I did this, this is so embarassing. Of *course* the sequences of consecutive primes on a leash will grow fast, and arbitrarily long. I thank
Message 3 of 11 , Mar 13, 2006
I can't believe I did this, this is so embarassing. Of *course* the
sequences of consecutive primes on a leash will grow fast, and
arbitrarily long. I thank you all for not calling 911 on my behalf.

in recovery from a brain bubble,
contritely,
Mark

<mark.underwood@...> wrote:
>
> Talk about another jump. After (159390,40) the next increases are
> (2106720,43) and (3573570,53)!
>

(snip)
• ... c.f. Lucky numbers perhaps? It may be clear something grows, but that doesn t mean that finding extremal values isn t interesting. c.f. prime gaps. Phil ()
Message 4 of 11 , Mar 13, 2006
--- Mark Underwood <mark.underwood@...> wrote:
> I can't believe I did this, this is so embarassing. Of *course* the
> sequences of consecutive primes on a leash will grow fast, and
> arbitrarily long. I thank you all for not calling 911 on my behalf.
>
> in recovery from a brain bubble,
> contritely,
> Mark

c.f. Lucky numbers perhaps?

It may be clear something grows, but that doesn't mean that finding extremal
values isn't interesting.

c.f. prime gaps.

Phil

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• Thank you Phil. I ve recovered enough to offer an addendum: The 9 primes after 1260 are 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47). I ve checked up to six
Message 5 of 11 , Mar 13, 2006
Thank you Phil. I've recovered enough to offer an addendum:

The 9 primes after 1260 are

1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).

I've checked up to six million and thus far the 9 string above is the
longest cluster of, for lack of a better term, doubly consecutive
primes.

Mark
• ... Jens will find you ones up to length 13 trivially, won t you, Jens ;-) Phil () ASCII ribbon campaign () Hopeless ribbon campaign / against
Message 6 of 11 , Mar 14, 2006
--- Mark Underwood <mark.underwood@...> wrote:
> Thank you Phil. I've recovered enough to offer an addendum:
>
> The 9 primes after 1260 are
>
> 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
>
> I've checked up to six million and thus far the 9 string above is the
> longest cluster of, for lack of a better term, doubly consecutive
> primes.

Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

Phil

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• ... Jens will find you ones up to length 13 trivially, won t you, Jens ;-) Phil () ASCII ribbon campaign () Hopeless ribbon campaign / against
Message 7 of 11 , Mar 14, 2006
--- Mark Underwood <mark.underwood@...> wrote:
> Thank you Phil. I've recovered enough to offer an addendum:
>
> The 9 primes after 1260 are
>
> 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
>
> I've checked up to six million and thus far the 9 string above is the
> longest cluster of, for lack of a better term, doubly consecutive
> primes.

Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

Phil

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• ... Here is 18: 1906230835046648293290030 + (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) But I didn t find it:
Message 8 of 11 , Mar 14, 2006
Phil wrote:
> Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

Here is 18:
1906230835046648293290030 + (13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83)

But I didn't find it: http://www.ltkz.demon.co.uk/ktuplets.htm#largest18
Joerg Waldvogel & Peter Leikauf report it's minimal for 13-83.

There are probably smaller 18's for sequences spanning more than 71 numbers.

--
Jens Kruse Andersen
• ... I have found 3 cases of 19 doubly consecutive primes . I think they are the first known above 18. n=66916817613475787 and n=304555583497054847 both give
Message 9 of 11 , Mar 17, 2006
Mark Underwood wrote:

> The 9 primes after 1260 are
>
> 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
>
> I've checked up to six million and thus far the 9 string
> above is the longest cluster of, for lack of a better
> term, doubly consecutive primes.

I have found 3 cases of 19 "doubly consecutive primes".
I think they are the first known above 18.

n=66916817613475787 and n=304555583497054847 both give consecutive primes:
n + (0,2,6,12,20,62,66,126,132,140,144,146,150,152,180,192,210,242,276)

n=149888370336254401 and n=224561886371432461 both give:
n + (0,12,18,36,76,112,126,148,186,196,208,222,246,250,252,256,258,298,316)

n=56029782174606241 and n=267322720146562081 both give:
n + (0,28,58,66,70,96,100,102,106,108,130,148,156,162,240,276,312,336,378)

The same 3 in Mark's notation:
237638765883579060 + (66916817613475787, ..., 66916817613476063)
74673516035178060 + (149888370336254401, ..., 149888370336254717)
211292937971955840 + (56029782174606241, ..., 56029782174606619)

q=34837285 prp quintuplets were computed on the form:
k*17# + (97, 101, 103, 107, 109), for k<734445220000

This ensured q*(q-1)/2 ~= 6*10^14 pairs with at least
5 consecutive primes in the same pattern.
The use of 17# gave the same divisibility properties
for primes<=17 to increase odds of more matches.

The previous 14 and next 14 prp's were computed for each quintuplet.
Comparison gave 3 cases with 14 combined consecutive matches.

The GMP library made prp tests. PARI/GP proved the solution primes.
The DOS/Windows command "sort" was used on a 1.6 GB file during comparison.
(Has "sort" ever been credited in a prime record?)

--
Jens Kruse Andersen
• Now this is amazing - not only are the primes doubly consecutive in clusters of 19 (!) but they each contain the same minimal quintuple. Brilliant, thank you
Message 10 of 11 , Mar 17, 2006
Now this is amazing - not only are the primes doubly consecutive in
clusters of 19 (!) but they each contain the same minimal
quintuple. Brilliant, thank you Jens!

Mark

--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
> Mark Underwood wrote:
>
> > The 9 primes after 1260 are
> >
> > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
> >
> > I've checked up to six million and thus far the 9 string
> > above is the longest cluster of, for lack of a better
> > term, doubly consecutive primes.
>
> I have found 3 cases of 19 "doubly consecutive primes".
> I think they are the first known above 18.
>
> n=66916817613475787 and n=304555583497054847 both give consecutive
primes:
> n +
(0,2,6,12,20,62,66,126,132,140,144,146,150,152,180,192,210,242,276)
>
> n=149888370336254401 and n=224561886371432461 both give:
> n +
(0,12,18,36,76,112,126,148,186,196,208,222,246,250,252,256,258,298,316
)
>
> n=56029782174606241 and n=267322720146562081 both give:
> n +
(0,28,58,66,70,96,100,102,106,108,130,148,156,162,240,276,312,336,378)
>
> The same 3 in Mark's notation:
> 237638765883579060 + (66916817613475787, ..., 66916817613476063)
> 74673516035178060 + (149888370336254401, ..., 149888370336254717)
> 211292937971955840 + (56029782174606241, ..., 56029782174606619)
>
> q=34837285 prp quintuplets were computed on the form:
> k*17# + (97, 101, 103, 107, 109), for k<734445220000
>
> This ensured q*(q-1)/2 ~= 6*10^14 pairs with at least
> 5 consecutive primes in the same pattern.
> The use of 17# gave the same divisibility properties
> for primes<=17 to increase odds of more matches.
>
> The previous 14 and next 14 prp's were computed for each quintuplet.
> Comparison gave 3 cases with 14 combined consecutive matches.
>
> The GMP library made prp tests. PARI/GP proved the solution primes.
> The DOS/Windows command "sort" was used on a 1.6 GB file during
comparison.
> (Has "sort" ever been credited in a prime record?)
>
> --
> Jens Kruse Andersen
>
• ... I computed 47315761 more quintuplets to search for 20 doubly consecutive primes . There wasn t any - but there was 21! n=446863043340173267 and
Message 11 of 11 , Mar 21, 2006
Mark Underwood wrote:
> The 9 primes after 1260 are
> 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).

I wrote:
> I have found 3 cases of 19 "doubly consecutive primes".
> 34837285 prp quintuplets were computed

I computed 47315761 more quintuplets to search for 20 "doubly
consecutive primes". There wasn't any - but there was 21!

n=446863043340173267 and n=534402442999154537 both give consecutive primes:
n + (0,2,56,62,80,110,146,150,152,170,230,234,252,264,276,
290,294,296,300,302,344)

The same in Mark's notation:
87539399658981270 + (446863043340173267, ..., 446863043340173611)

GMP prp'ed and PARI/GP proved. This time "sort" worked on a 3.4 GB file.
I had to delete things to get room. The search has stopped.

--
Jens Kruse Andersen
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