About prime and semi-prime tuples I wrote:
> Not exactly the same complexity with the right algorithm
David Broadhurst wrote:
> Incidentally the real breakthrough for the prime triplet record is 4135
> digits was avoiding ECPP. But this was still an O(digits^5) problem,
> whereas yours was O(digits^4).
"Not exactly" was a deliberate understatement. I know there is a huge
difference and it makes little sense to compare sizes, except for fun. The
"wrong" algorithm for k consecutive numbers with the same number of prime
factors (used for a 7-tuple with 3 prime factors by Phil, who admitted to
being sloppy) is: Search k simultaneous prime cofactors, e.g. x/5, (x+1)/2,
(x+2)/3. This was also the suggestion of others in alt.math.recreational and
would give the normal prime tuple complexity O(k+2).
David Broadhurst wrote:
> PS: I re-read saw that you did your own trial factoring, sorry.
> But I guess that it was not truly a sieve, but just for
> one pair at a time? Mind you it's not easy to see a way
> round that. If anyone could sieve this two-parameter
> input problem, it would be Phil.
Yes, I trial factored one pair at a time. My candidates were on the form
p*q = (c*6*4691#+1) * (d*6*4691#+1)
This (including the 6) prevents factors below 4691 in both (pq+1)/2 and
I precomputed an array of 6*4691# mod (primes<10^8).
Then I could quickly trial factor each pair from 4691 to 10^8 (a high limit
for individual trial on 4k-digits with a 2.7s prp test) using only 32-bit
inline assembler instructions on c and d.
I thought about sieve possibilities but it seemed hard and with a few days
expectation, I dropped it. I would have given the 10k-digit problem more
I should also admit that I was a lot more lazy for p = (c*6*4691#+1). This is
a good sieve form and it turned out I needed 1314 primes (bad luck), but I
used pfgw's trial factoring for this. Later I discovered the NewPGen check box
"use primorial mode" in plain sight. I had only looked through the type
selection box... When it opens it actually covers the primorial box and I
thought there was only k*b^n something available. Why didn't Paul Jobling put
a hint in the type box for people like me with tunnel vision :-)
Never underestimate the potential for stupidity among software users.
Phil Carmody wrote:
> In the world of sieving, Jens is one of the people to whom I doff my hat.
> He stood out on alt.math.recreational because of his sieving ability.
Thanks. However alt.math.recreational is not hard competition, at least before
you joined it. You are the better siever and I have only beaten your speed a
couple of times (including the prime puzzles) by spending much more time on
special-purpose programs. GenSv seems impressive for such a generic sieve.
Jens Kruse Andersen