## Re: [PrimeNumbers] three-prime sum chains

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• ... With the choice we quickly get a chain with 12 sums: 281 + 283 + 293 = 857 857 + 859 + 863 = 2579 2551 + 2557 + 2579 = 7687 7681 + 7687 + 7691 = 23059
Message 1 of 7 , Jun 4, 2013
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Phil Carmody wrote:
> I specifically wanted to offer the choice. I wanted the chains
> to be longer, and also less simple to find.

With the choice we quickly get a chain with 12 sums:
281 + 283 + 293 = 857
857 + 859 + 863 = 2579
2551 + 2557 + 2579 = 7687
7681 + 7687 + 7691 = 23059
23059 + 23063 + 23071 = 69193
69193 + 69197 + 69203 = 207593
207569 + 207589 + 207593 = 622751
622751 + 622777 + 622781 = 1868309
1868287 + 1868291 + 1868309 = 5604887
5604881 + 5604887 + 5604901 = 16814669
16814669 + 16814671 + 16814701 = 50444041
50444027 + 50444041 + 50444059 = 151332127

The longest chain with initial primes below 10^11 is probably 14 sums:
2878090951 + 2878090961 + 2878090967 = 8634272879
8634272839 + 8634272879 + 8634272891 = 25902818609
25902818609 + 25902818629 + 25902818663 = 77708455901
77708455819 + 77708455823 + 77708455901 = 233125367543
233125367543 + 233125367599 + 233125367609 = 699376102751
699376102751 + 699376102757 + 699376102783 = 2098128308291
2098128308239 + 2098128308273 + 2098128308291 = 6294384924803
6294384924787 + 6294384924803 + 6294384924823 = 18883154774413
18883154774413 + 18883154774441 + 18883154774447 = 56649464323301
56649464323301 + 56649464323303 + 56649464323307 = 169948392969911
169948392969877 + 169948392969881 + 169948392969911 = 509845178909669
509845178909611 + 509845178909669 + 509845178909687 = 1529535536728967
1529535536728967 + 1529535536728987 + 1529535536728999 = 4588606610186953
4588606610186927 + 4588606610186947 + 4588606610186953 = 13765819830560827

There are also 14 starting with
44453980303 + 44453980309 + 44453980379 = 133361940991

> I presume you just ran a prime sieve and then probed for each one
> with no deduplication?

I didn't bother with deduplication but I ran two prime sieves in
parallel, around p and 3p. This meant the first sum never had to be prp
tested, and its surrounding primes were also generated without prp tests.
It could be extended to a third sieve around 9p and so on, but each
extra sieve has to sieve a larger interval and avoids fewer prp tests.

> I hadn't considered Dickson. It's the big gun, certainly.
> I wondered initially if it would fail on guaranteeing that the
> positioned primes are consecutive, but if you ask for a triplet

Dickson's conjecture can be manipulated to ensure consecutive primes
in this problem for any admissible pattern. We can just demand extra
primes chosen so specific prime factors are forced to divide certain
numbers in a gap.
For example, we can ensure consecutive {p, p+6} by demanding four primes
{p, p+6, p+8, p+14}. This forces 3 to divide p+4, and 5 to divide p+2.

--
Jens Kruse Andersen
• ... which is a special case of Bateman-Horn: http://tech.groups.yahoo.com/group/primenumbers/message/25133 David
Message 2 of 7 , Jun 4, 2013
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:

> Arbitrarily long chains are certainly expected.
> It would for example follow from Dickson's conjecture:
> http://primes.utm.edu/glossary/xpage/DicksonsConjecture.html

which is a special case of Bateman-Horn: