Primes in π

[kar_bon]

Originally Posted by davar55

Also, where are the first occurrences of the Mersenne prime exponents.
(The 8 digit ones may be far to find.)

Searched the first 1,000,000,000 digits of PI to find this (leading '3' not counted):

Mers Expo	start in PI at digit
2 		6
3 		9
5 		4
7 		13
13 		110
17 		95
19 		37
31 		137
61 		219
89 		11
107 		1487
127 		297
521 		172
607 		286
1279 		11307
2203 		1910
2281 		19456
3217 		959
4253 		7337
4423 		7591
9689 		690
9941 		1073
11213 		47802
19937 		115211
21701 		28507
23209 		280538
44497 		85342
86243 		89373
110503		808004
132049 		840293
216091 		3226144
756839 		996061
859433 		2887812
1257787 	24078017
1398269 	2037623
2976221 	20104152
3021377 	1220576
6972593 	9252419
13466917 	39603620
20996011 	40909479
24036583 	8854005 
25964951 	19456503
30402457 	645842094
32582657 	510029176
37156667 	53909580
42643801 	228338527
43112609 	248103197

Curious:
Mersenne expo 127 starts at index 297 which is 129₁₆.

[Xyzzy]

What is the largest known prime in the sequence of digits of pi?

[science_man_88]

Originally Posted by Xyzzy

What is the largest known prime in the sequence of digits of pi?

http://oeis.org/A060421 supposedly shows that one known one is up to over 78000 digits depending on how you define a pi prime

[davar55]

Originally Posted by science_man_88

http://oeis.org/A060421 supposedly shows that one known one is up to over 78000 digits depending on how you define a pi prime

It said 78073. I think it's remarkable that batalov's computations
are going to or have already exceeded that.

[Batalov]

I've half-heartedly tried to check the same run up from 78073 to 100k (and the a(20) and a(96) to 100k) - no primes (and then the run gets slow), so prp78073 holds the palm d'or as far as we know. It can be easily beaten with random starts and in the range of lengths from 78074 to 80-85k, but that would be fairly pointless -- that in turn would be easily beaten.

[davar55]

Originally Posted by Batalov

I've half-heartedly tried to check the same run up from 78073 to 100k (and the a(20) and a(96) to 100k) - no primes (and then the run gets slow), so prp78073 holds the palm d'or as far as we know. It can be easily beaten with random starts and in the range of lengths from 78074 to 80-85k, but that would be fairly pointless -- that in turn would be easily beaten.

The OP did suggest the sequence from 1 to 100 ...

(Don't want to light any fires, but breaking records is always fun.)

[davar55]

Originally Posted by Batalov

I've half-heartedly tried to check the same run up from 78073 to 100k (and the a(20) and a(96) to 100k) - no primes (and then the run gets slow), so prp78073 holds the palm d'or as far as we know. It can be easily beaten with random starts and in the range of lengths from 78074 to 80-85k, but that would be fairly pointless -- that in turn would be easily beaten.

How about "Our sequence can beat your sequence" as a humorous
motivation? I would love to know the length of the values for 20 amd 96.

[Batalov]

They are longer than 103,000 digits. :-)

[davar55]

Originally Posted by Batalov

They are longer than 103,000 digits. :-)

Love that emoticon. I do believe there's something up your sleeve .....

[ixfd64]

Because pi has an infinite number of digits, it's almost certain that every possible sequence can be found. I wonder how far one will have to go in order to find, say, M#47?

[Batalov]

Well, ok, records are made to be broken. With a bit of luck I found a 140,165-digit PRP that starts with the first "96" in Pi, the a(96). This may also be the largest known PRP in the sequence of digits of Pi, for Xyzzy.

I am DCing have doublechecked it in a few bases and with combined N+1/N-1 and submitted to Lifchitz. Here's the code to generate the number for the independent checks:

# Pari/GP #
\p 143000
prp=floor(Pi*10^140344)%10^140165;
# passes the GP ispseudoprime(prp) test, too, in addition to PFGW-based PRP and BLS

a(20) is still ongoing.

EDIT2: strictly speaking, because a(96) is quite big - it may not be a minimal solution: there's a chance that by way of some bug I could have missed some smaller PRP (I also have a small gap between two threads that processed candidates above and below 125,000 digits, which I will close sometime soon; I may re-run the whole search using a different base for PRP, too -- or anyone else is welcome to. The scripts are all here, in this thread.)