Octoproths

[axn]

Results for n = 130, (k < 10^13)

1075252753275
3408331609305
7076113724805

[Dougy]

I've been looking at the small bases. (primes, rather than probable primes) I wrote my own program to look at these.

There are no octoproths with base n = 26 or below.

The first one is
109989075 27
and is the only one with base 27.

The next are
21207165 28
191093475 28
are the only two with base 28.

...more to come

One interesting one is n=1, k=15.
15*2^1+1 = 31
15*2^1-1 = 29
15*2^(1+1)+1 = 61
15*2^(1+1)-1 = 59
2^1+15 = 17
2^1-15 = -13
2^(1+1)+15 = 19
2^(1+1)-15 = -11

If you count negative primes too.

[robert44444uk]

Dougy

I am really surprised that there are no "small" octos. The way I have defined them means that negative numbers, created through the 2^n-k calculation, rule that number out, so your interesting case has to remain as that.

But thank you for looking at the small case. I just find the result hard to believe, but the negative rule counts out a lot for small n, especially when k goes in multiples of 15 (almost 2^4), so maybe I should have realised.

Maybe you should post the full decimal value of this find to Chris Caldwell's Prime curios page:

http://primes.utm.edu/curios/

Regards

Robert Smith

[robert44444uk]

Dougy

I just realised, (as I am sure you have) that you will need also to look at higher n, because they may have a smaller k value, such that k.2^n+1 is a smaller number.

So that you will have to check almost all the way up to n=50 to make totally sure there are no smaller octos.

Regards

Robert

[Dougy]

So, if my program works properly, there are no (certified prime) octoproths within the ranges n=31-50 and k=15-21207165.

Furthermore
328724235 29
233752995 30
are the only octoproths with those bases.

So this is a proof that
21207165*2^28+1 = 5692755007242241.
109989075*2^27+1 = 14762483751321601.
are the smallest two octoproths.

Also 21207165 is also the smallest known k-value forming a octoproth. I wonder if it's actually the smallest possible. I might search with a fixed k and varying n instead. (but that'd require writing a whole new program)

It would be nice if someone could verify this independently before I submit it anywhere.

I working on the new version now...

[Dougy]

If k*2^n+1 is an octoproth then

k = 1 mod 2. If k is even then 2^(n+1) + k is divisible by 2.

k = 0 mod 3. If k = 1 mod 3 then either 3 divides k*2^n+1 or k*2^(n+1)+1. Similarly for k = 2 mod 3

k = 0 mod 5.

k = 0 mod 7 or (n = 1 mod 3 and k = +/- 1 mod 7).

I can't make any other useful criteria from any other primes. Does anybody know of other goodies like this?

[Dougy]

Only two octoproths for n=31, this base is now complete.
196168335 31
1813059975 31

[axn]

Originally Posted by Dougy

So, if my program works properly, there are no (certified prime) octoproths within the ranges n=31-50 and k=15-21207165.

Furthermore
328724235 29
233752995 30
are the only octoproths with those bases.

So this is a proof that
21207165*2^28+1 = 5692755007242241.
109989075*2^27+1 = 14762483751321601.
are the smallest two octoproths.

Also 21207165 is also the smallest known k-value forming a octoproth. I wonder if it's actually the smallest possible. I might search with a fixed k and varying n instead. (but that'd require writing a whole new program)

It would be nice if someone could verify this independently before I submit it anywhere.

I have verified that there are no octo's between 10 <= n <= 26. Also there are no octo's in the range 31-50 for k < 10^7. I am right now in the process of checking whether 21207165 is the smallest possible for n <= 1000

[Dougy]

Today I took a look at the number of candidates remaining after running axn1's sieve to 10^10. I ran the sieve over n=50 to n=150.

I will call the "weight" of a base n, to be the number of candidates remaining after running the sieve through 10^10.

The number of candidates remaining:
Average weight = 18.15
Minimum weight = 3 (n=68) the lightest.
Maximum weight = 54 (n=112) the heaviest.

Also
8299358445 50
3920165865 54
7130617935 62
925905105 64
3539387145 65
were the only prime-octoproths found.

I've attached an excel spreadsheet with the details, and 101 text files with the output from the sieve for n=50 to 100.

[Dougy]

Firstly it seems that the 'heavy' bases are more likely to produce an octo than the 'light' bases. So n=52, 67, 70, 82, 97, 112, 115, 142, ... (bases with weight >= 40) would be a good place to start searching. With this reasoning (whether sound or not) I discovered:
65498827395 67

...

In an attempt to make this search a bit more organised I've created a text file listing what has already been searched. I'll try to update this regularly, and as often as possible.
Btw, tell me if I've missed anything or you've searched a region more than what is listed.

In this file, a typical base would look like this:

<k> <n> <discoverer>
<k> <n> <discoverer>
...
<k> <n> <discoverer> (searched/2^n)

if an octo exists, and

<n> (searched/2^n)

otherwise. This way we can hopefully not redo others work.

PS: If I haven't missed any, we now know of 97 octoproth-primes. Hopefully we'll make the 100 mark soon.

PS2: Most wanted octoproths-primes: base 32, 63.