Primes in π

[LaurV]

Nice work. However, extending the sequence to higher starters is not big deal, as the most of them are "easy".

Both "with initial 3" and "without" were discussed before, this does not change the things too much, so we decided to let the 3 out of it. Also, the problem asks for primes which "extends" the initial starting number. There is not so much fun to say that the prime starting with 3 is 3, or the one starting with 17 is 17. (see post #45, onwards).

The only interesting case remaining after Batalov's work is actually "20". Who can solve the 20 gets a bonus...

Beside of it, the 10, 17, 80, 81, 84, 96 need to be proved prime in factorDB (they are only PRP; the 54, 62, 73, 97 were already proved prime). This is secondary.

The main goal remains 20.

[danaj]

Originally Posted by LaurV

Nice work. However, extending the sequence to higher starters is not big deal, as the most of them are "easy".

But there are 8 more in his results that are indicated at 170k+ digits or more, so not all are easy.

Both "with initial 3" and "without" were discussed before, this does not change the things too much, so we decided to let the 3 out of it.

The scripts and discussion on page 3 use the initial 3. davar's post on page 3 indicates "Indexing could begin with the 3 as 1 or 0" meaning the 3 was intended to count. davar's post on page 5 also indicates 3 is included. Batalov's script on page 5 Batalov's script isn't entirely clear, but if the output file is untouched it has "3.14..." in it meaning it wouldn't catch the initial 3 because of the decimal point left in. Your results on page 5 leave them out.

I don't think this was discussed. People seem to have just picked something and sometimes mention what they chose. The original poster twice indicated the 3 should be included.

I think it's a terrible sequence if we have 4 different versions.

Also, the problem asks for primes which "extends" the initial starting number. There is not so much fun to say that the prime starting with 3 is 3, or the one starting with 17 is 17. (see post #45, onwards).

The original post is misleading in that it says find the number then find "the first prime constructed from the subsequent digits". So for 1 you should find the 1, then choose "41" since that is the first prime made from subsequent digits. But we all know that wasn't what was meant. We want the prime made from the first occurrence of the number concatenated with the fewest [positive] number of subsequent digits. Without "positive" we would accept 0 additional digits, otherwise not. The "uninteresting" bit came from post 29 rather than 45.

What I meant by saying that if we extend the series this becomes less important, is that sure, we find "7" for 7, but go a little farther and we'll find each 7x, and later 7xx, etc. I don't particularly care which one is chosen, but it would be nice if we all actually worked on the same sequence.

The only interesting case remaining after Batalov's work is actually "20". Who can solve the 20 gets a bonus...

The only interesting case up to 100, you mean. Did you look at JF's post? 20 would certainly be nice to solve. If only I had more spare computers :)

I got most of what I wanted from it -- a nice speedup of my pretest for large (50k+ digit) numbers. I'll probably run a(20) farther later, but someone else will likely beat me to it. I can run primo on 17, 81, and 84, but 10, 80, and 96 look daunting.

[LaurV]

Grrr... I have to write you down in my book under the chapter "people not to argue with, ever!".

(that was a compliment, told with much respect. don't get fussy about it!)

[danaj]

:( Sorry.

[J F]

Danaj, errors corrected, thanks!

The list in the first post doesn't include the leading 3,
and for 2 and 3 results are equal to the number, so I
just used this ruleset. Appending digits to prime starting
numbers or not doesn't really matter, the information is
still there. Example: result for a(2) is either 2, or the
same als a(26). No need to calculate anything new, just
rearrange the list.

[kar_bon]

The value "31415926" occurs at position 50366471 ("3." not counted) of pi.
No longer value of this pi-like number in the first 1e9 decimal digits of pi.

[danaj]

"314159265" occurs at position 1660042750. I didn't find the next value in the first 5000M digits.

The current list in the first post may or may not include the first 3 -- the only number starting with 3 listed is "3" with no starting position noted. It looks like almost all the OEIS pi/prime related series include the beginning 3, e.g. Pi-Prime (OEIS A005042) and all the crossref'd entries it has.

I agree with you (JF) on the uninteresting bit. At first I thought it was a good change, but now I think it just adds complication that doesn't really add value. As you point out if a(2) is boring, just go to a(26) for the interesting part. If a(41) is boring, look at a(415).

[J F]

some Update: Increasing run time and decreasing chance
per PRP-test have dampened my enthusiasm a bit, so I
did cut back from 3-4 cores nearly 24/7 to 2 cores
part-time. #20 passed 356K digits, still no luck.
A friend is donating 1 core part-time working on #196,
passed 250K digits. The other 6 unsolved up to 1111
(#380, #422, #861, #899, #955 and #988) are brought
to 200K digits and parked for now.

Topic leading 3 or not: for most starting values this
will do nothing except shifting their offset by 1.
The others will lead to OEIS A005042.
Again, not really information gained/lost.

[ewmayer]

Originally Posted by bsquared

IANANT, but the infinite sum of 1/ln(n) diverges, so even accounting for the fact that on average we only sum 4 of every 10 terms, I would think that the probability that there exists a prime would be 1.

This would also appear to follow if the digits are normal, which is generally believed but as yet unproven.

If normality holds, the digits will further be "as random as can be", but note the above 2 properties will be true of all normal reals, which are (provably) a dense subset of the reals. Interestingly, the normals likely include both irrationals and transcendentals - for example, sqrt(2), pi and e are all generally believed (but not proven) to be normal.

I find it interesting that is far easier to prove that almost all reals are normal than to prove that a selected one, even one as well-studied as sqrt(2), is.

[davar55]

Just wondering if any progress has been made in this series,
especially a(20)?

[J F]

a(20) neares 450k digits, still nothing
a(196) finished with a 312306-digit PRP