Primes in π

[davar55]

It's a kinda cool race which I'm enjoying from
the grandstands of my home PC (HP 64-bit Windows 10).

What's the ET2CU (estimated time to catch up) for the OP a(20) now past 500k
versus
pi(0,314): 314159265358979323846...89830932080370010789 (613373 digits)
Lucas PRP! (70233.3140s+30.6992s)
??

[davar55]

No activity here in a while. Has a(20) reached a milestone yet?
When it reaches 700000 (or 1000000) I think it would be cause
to pause for some kind of NT analysis...is it seriously possible
for the a(20) series to have NO primes...if so, is this a unique
instance in pi...are there other related implications.

[xilman]

This thread was started so long ago that I had to review its start to understand what's going on. I came across the following:

Originally Posted by Batalov

That's not what I meant. The inner loop is already done to death. You simply cannot use the outer loop (as the problem is stated).

Suppose we are searching for a(20).

pi = 3.
14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196...

Suppose we've checked the substrings starting from the red 20 up to a length of million and didn't find a prime. That doesn't give us the right to move on to the blue 20, find 2089 and say that we are done.
We can only move on to the next instance of the starting point after we have proven that there are no primes formed by the first instance. Can we prove that?

It is a famous unsolved problem as to whether pi is normal in all bases or, indeed, to any base. It is widely believed (I believe!) that it is normal in all bases. That being the case, it is normal in base 10.

To expand a little: normality to base a means that all possible strings containing the digits of a occur somewhere in the base-a expansion of pi. In particular, all possible decimal primes are hypothesized to be contained somewhere in the decimal expansion of pi. The problem is finding them.

I'm sure that Serge's analysis is correct. I haven't yet proved under the normality conjecture that his red sequence does NOT contain a prime if continued far enough.

[rogue]

Originally Posted by xilman

In particular, all possible decimal primes are hypothesized to be contained somewhere in the decimal expansion of pi. The problem is finding them.

Is there a project searching for all primes in the decimal expansion of pi? In other words does someone know the smallest prime that is not found in that decimal expansion?

I'm sure that Serge's analysis is correct. I haven't yet proved under the normality conjecture that his red sequence does NOT contain a prime if continued far enough.

How does one go about proving that? It doesn't seem provable.

[xilman]

Originally Posted by rogue

Is there a project searching for all primes in the decimal expansion of pi? In other words does someone know the smallest prime that is not found in that decimal expansion?



How does one go about proving that? It doesn't seem provable.

Are ytou suggesting that it's an undecidable proposition in the Gödelian sense?

I really have no idea whether the proposition is true, false or undecidable.

[J F]

With the help of pixsieve (thanks again, rogue!) and
a new 6600K its now 5-7x faster than my old machine.
Enthusiasm rekindled, chugging along.
#20 is approaching 600K and the other 8 unfinished
up to 1666 are between 250K and 300K.

[J F]

First hit with the new machinery.
#861, 279430 digits PRP.

[ATH]

Originally Posted by Batalov

Incidentally, an interesting entry was added to PRPtop last month.
PIPrime(613373) (613373 digits, ...obviously) -- Adrian Bondrescu 05/2016

Originally Posted by CRGreathouse

Nice find! Have all the smaller ones been cleared?

I checked all the way up to this prime and up to 650K without finding any other primes.


Here are factors up to 1,000,000 digits sieved to 134G (including a few ECM factors):
pifactors.txt

Ignoring all the even numbers and those ending in 5 and removing these factors and known primes leaves these candidates (I didn't remove the new 613373 digit prime):
remaining.txt

Here are pfgw logs from running these up to 650K, use http://7-zip.org/ to unpack them even though they are called ".zip".
000k-100k.zip (139 KB, unpacks to 104 MB)
100k-200k.zip (297 KB, unpacks to 316 MB)
200k-300k.zip (532 KB, unpacks to 540 MB)
300k-350k.zip (389 KB, unpacks to 342 MB)
350k-400k.zip (439 KB, unpacks to 399 MB)
400k-450k.zip (487 KB, unpacks to 449 MB)
450k-500k.zip (538 KB, unpacks to 512 MB)
500k-550k.zip (823 KB, unpacks to 567 MB)
550k-600k.zip (889 KB, unpacks to 609 MB)
600k-650k.zip (965 KB, unpacks to 702 MB)

[davar55]

A large prime in the digits of pi is interesting, although except for the unusual
magnitude I don't see right off how it may be number-theoretically important.
Perhaps there's a
finite significant collection of out-of-proportionally-large such primes whose numeric
values combined in some way might lead to a theorem? Musing...
If there had been an actual prime gap at a(20), and I don't say how that might have
been proven, I think that might have been more significant, since the number of
such prime gaps might be finite.

[rogue]

Originally Posted by davar55

A large prime in the digits of pi is interesting, although except for the unusual
magnitude I don't see right off how it may be number-theoretically important.

Aren't most primes that people search for unimportant? Maybe someone should start a "why I search for primes" thread.

[VBCurtis]

perhaps davar55 ought to define "important" from his viewpoint, so we can know which projects he finds unimportant.