Octoproths

[Greenbank]

Originally Posted by R. Gerbicz

For this I hope you use kmin=1 and kmax=2^n-1

But also note that in this case the second smallest number 2^(n+1)-k>2^n>10^5, but the smallest number is 2^n-k<10^5 if and only if 2^n-1>k>2^n-10^5 ( because here 2^n-1>k ). It is important because my sieve will delete this prime from the list, because it has got a small prime factor (<10^5). It means that my program miss all octoproth in [2^n-10^5,2^n-2] range and you have to check it by another program ( it isn't very hard because it means you have to check at most 8*10^5 numbers for prp testing ).

And yes it is very important that after you have gotten your text file then check by another program the results, that these are really prime numbers. I think for n>50 it isn't very impossible to get a composite 3-prp numbers.

On page2 there were some results up to n=43 or something like that, have you checked these results, by my octo program? It would be very good.

It isn't impossible to write a faster program for these small n values, because prp testing is much-much faster than sieving for these small n values.

Yes, I'm going from 1 to 2^n.

I see what you mean about the top end of a range, I'll make a note that these ranges need to be rechecked when we have something that can check them.

So far every 3-prp number your program has output has been confirmed as prime. However I will always check them with the PARI script to be sure.

As for double checking, that is something to bear in mind. Again, I'll add that to my list and have a think about it.

P.S. Results for completed ranges posted in a new thread, with downloads.

[robert44444uk]

I rather like the idea of looking for spiders, with between 0 and 8 legs.

Take an octoproth, this will form the body, with four segments.

k*2^n+1,k*2^(n+1)+1
k*2^n-1, k*2^(n+1)-1
2^n+k, 2^(n+1)+k
2^n-k, 2^(n+1)-k

Now see how many legs the spider has. Each segment has two possible legs, representing n values one less and one more than on the segment. For example:

Possible left leg Body Possible right leg
k*2^(n-1)+1 k*2^n+1, k*2^(n+1)+1 k*2^(n+2)+1

The possible legs become legs when they are prime.

A dodecaproth with have four left or 4 right legs, a fully fledged spider will have eight legs.

Given that these forms are likely to be highly rare, it would be interesting to see the distribution of legs on those existing octos already discovered. It is highly probable that we will find octos with 4 legs, although they will not all be on one side. 5 legs would be nice. 6 a bonus although unlucky.

Regards

Robert Smith

[Greenbank]

n=51 is complete and has 16870 confirmed Octoproths. Will add this to the info and downloads shortly.

[smh]

Originally Posted by Greenbank

The first 4 were the ones found with axn1's Pascal program which let a few more pseudo-primes through than octo.exe.

This is not true. axn1's program worked fine, it was just Templus who forgot to extend the ABC file to test for the remaining 4 cases.

See post 65 - 69 of this tread.

[Greenbank]

Originally Posted by smh

This is not true. axn1's program worked fine, it was just Templus who forgot to extend the ABC file to test for the remaining 4 cases.

See post 65 - 69 of this tread.

We're both right. ;-)

axn1's program still outputs those values, otherwise Templus would not know them to pass them through PFGW. octo 4.5 (and previous versions) does not even output these values as possible octoproths.

And yes, Templus forgot to add the last 4 cases to the ABC line.

[R. Gerbicz]

Greenbank can you modify the function of f=floor() to f=round(), see the number of octoproth thread, because in this case it'll give also that for n<27: f(n)=0 and for n>=27 : f(n)>0
ps in some cases it'll give the correct number of octoproths!!!

[axn]

Originally Posted by R. Gerbicz

But also note that in this case the second smallest number 2^(n+1)-k>2^n>10^5, but the smallest number is 2^n-k<10^5 if and only if 2^n-1>k>2^n-10^5 ( because here 2^n-1>k ). It is important because my sieve will delete this prime from the list, because it has got a small prime factor (<10^5). It means that my program miss all octoproth in [2^n-10^5,2^n-2] range and you have to check it by another program ( it isn't very hard because it means you have to check at most 8*10^5 numbers for prp testing ).

Please see this post earlier in the thread (at the bottom). I have already verified that there are no octos in [2^n-10^5,2^n-2] for n <= 64. What I did was, I modified the original sieve to count down from 2^n to 2^n-10^5 and disabled the code for sieving the 2^n-k form (so sieving with 7 forms instead of 8, which was still very good). I don't remember how many candidates survived the sieve, but there were no octos

[axn]

Originally Posted by Greenbank

We're both right. ;-)

axn1's program still outputs those values, otherwise Templus would not know them to pass them through PFGW. octo 4.5 (and previous versions) does not even output these values as possible octoproths.

And yes, Templus forgot to add the last 4 cases to the ABC line.

For the record, that program didn't do _any_ PRP tests It was only a sieve, meant to replace NewPGen. It was up to user to later do the PRP and find out which are the actual octos.

[R. Gerbicz]

Originally Posted by axn1

Please see this post earlier in the thread (at the bottom). I have already verified that there are no octos in [2^n-10^5,2^n-2] for n <= 64. What I did was, I modified the original sieve to count down from 2^n to 2^n-10^5 and disabled the code for sieving the 2^n-k form (so sieving with 7 forms instead of 8, which was still very good). I don't remember how many candidates survived the sieve, but there were no octos

I've checked that page, but : 3762658725,32 is still not an octoproth! I don't know but it is possible that your version has got a bug! ( for n=32 there are 6 octoproth )

[robert44444uk]

32 was the first number I checked, and I went up to k=4bn, using NewPgen as the initial sieve, and pfgw for primality testing, and only found 6, so that the 3762658725 value was not amongst those - see message #1 in this thread!!

Regards

Robert Smith

[axn]

Originally Posted by R. Gerbicz

I've checked that page, but : 3762658725,32 is still not an octoproth! I don't know but it is possible that your version has got a bug! ( for n=32 there are 6 octoproth )

Indeed! That shouldn't be there.

Blame it on human error. The primality testing was a manual process. I tried to be careful but somehow this one has slipped thru

Anyway, the important thing about that post was the last bit -- i.e. no Octos in [2^n-10^5,2^n-2] for n <= 64.