素数
Because these triples are a bit more prime than expected, an
interesting puzzle is to construct a pyramid structure as follows:
Take 5 consecutive primes, p,q,r,s,t
If A=p+q+r is prime, and B=q+r+s is prime, and C=r+s+t is prime, and
Z= A=B+C is also prime, then you have a sort of pyramid construction.
These are quite common, the first of which is
19,23,29,31,37
71,83,97
251
With 6 consecutive primes you can build along, but not up yet, a
flattened pyramid. The first of these is
2731,2741,2749,2753,2767,2777
8221,8243,8269,8297
24733,24809
With 7 consecutive primes, it is possible to go one layer higher, as
you would have 5 triplets in the second layer, and 3 sums of triplets
in the third layer, and 1 top layer, also prime, which is the sum of
the 3 sums in the third layer. Altogether that means (7)+5+3+1 primes,
or 9 numbers to be shown prime, equivalent to an 9-length cunningham
chain in terms of difficulty of finding.
Going higher is probably possible given the current power of
computers, and given ingenuity, as this is equivalent to 7+5+3+1 =16
primes to find.
A check of all primes up to 163315303 revealed
5 consecutive primes: 2409 pyramids
6 consecutive primes: 35 flattened pyramids
7 consecutive primes: 0
The first (7, 5, 3, 1) pyramid with 7 consecutive primes:
[67144843187, 67144843189, 67144843201, 67144843211, 67144843217,
67144843243, 67144843273]
[201434529577, 201434529601, 201434529629, 201434529671, 201434529733]
[604303588807, 604303588901, 604303589033]
[1812910766741]
The next starts at 76872991093. Search stopped at 2*10^11.
This is much harder than a CC9. It requires 10 simultaneous primes
(including one of the original 7) with no good way to sieve.
(9, 7, 5, 3, 1) seems extremely hard.
Robert wrote:
--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
> This is much harder than a CC9. It requires 10 simultaneous primes
> (including one of the original 7) with no good way to sieve.
> (9, 7, 5, 3, 1) seems extremely hard.
I can't see that it is equivalent to CC10, as by definition all the
members of the inital base of 7 primes are prime. In terms of time
taken perhaps, it may be that using "nextprime" in any program is
expensive.
One of the 7 primes must be included in a count of simultaneous primes.
Suppose we wanted 7 consecutive primes, and 9 other primes which formed
a CC10 with the original prime. This is obviously not like finding a CC9.
You are basically saying that because we start with 7 consecutive
primes instead of 1, it should count as 0.
Is there no mathematics which could underpin a sieve to tackle the 9-
pyramid? For example, the work you and others (especially Phil) did
recently showed, I think, that all of the primes which support a
pyramid member must not all be 1mod3 or all 2mod3. Because of the
interlocking nature of the pyramid it means that any permutations of
3 consecutive base members of the pyramid must meet this criterion,
which means, I think, a limited number of patterns.
Yes, there is a limited number of patterns mod 3. But this is in the
non-simultaneous consecutive primes.
Your considerations would imply to sieve for them as 5 simultaneous
primes in a pattern.
That would turn a (5, 3, 1) pyramid with 1+3+1 simultaneous primes
into a far harder 5+3+1.
In a potential (9, 7, 5, 3, 1) pyramid, that adds 8 simultaneous
primes to the problem.
Considerations about possible modular values of the consecutive
primes can only help in one way:
Make it slightly faster in some cases to determine whether already
computed consecutive primes might be a pyramid base.
Until the consecutive primes are computed, we don't know what
their modular values will be.
And if we fix the modular values before searching the primes, then we add a
bunch of simultaneous primes and make the problem far harder.
I already found two (7, 5, 3, 1) pyramids and (9, 7, 5, 3, 1) is far too hard,
so I'm not working more on this challenge.
The best algorithm may involve computing all primes or prp's until the first
solution. That might be around 25 digits and take billions of GHz years.
