P.I.E.S. - Prime Internet Eisenstein Search

[paulunderwood]

Originally Posted by pepi37

Is there end of your prime findings?

[axn]

What is the status of 49152? Is it nearing completion? 26 primes and counting...

[Batalov]

Yes, near finishing. (It was run on a few EC2 nodes and now I only have a clean up run closing small unfinished gaps; the largest being ~562k-600k.)

The slope of the sequence of b values is reasonably close to the ad hoc expected γ • (2m) ≈ 56721

[Batalov]

Originally Posted by Batalov

I've systematically scanned the Phi(3,-b^2^15) P.I.E.S. series and here is the chart for the numerologists to ponder.

This is the distribution of the 30 b values for which Phi(3,-b^2^15) is prime, and it is fairly clumpy in parts: there are seven primes for b<46,000; ...

There are now 48 of the Phi(3,-b^2^15) P.I.E.S. and the slope is remarkably close to γ • (2m) ≈ 37814

[Batalov]

While filling in missing values (a(11)-a(13)) in https://oeis.org/A298206 found a rather "late" bloomer prime (165394^49152-165394^24576+1, a 256501-digit prime).
a(13) = 165394 stands out quite far from the "expected range to find it" (~30,000).

Just an illustration for statistical inferences about "where the next Mersenne is" or the like.
It is where it is, no matter where it was "expected".

P.S. Left the scripts running for a few minutes and the next one showed up almost immediately: a'(13) = 165836.
165394^49152-165394^24576+1 is a 256501-digit prime.
165836^49152-165836^24576+1 is a 256558-digit prime.
No smaller b<165394 with b^49152-b^24576+1 prime.

That's even funnier: this is directly related to silliness of those who stop searches for certain "k" values in Riesel search project - just because "there was a prime found just now. Now there will not be another one for miles." No, silly, the probability is exactly the same for each next candidate!