[QUOTE=davar55;307534]Also, where are the first occurrences of the Mersenne prime exponents.
(The 8 digit ones may be far to find.)[/QUOTE]

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

[code]
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
[/code]

Curious:
Mersenne expo 127 starts at index 297 which is 129[sub]16[/sub].

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

[QUOTE=Xyzzy;307574]What is the largest known prime in the sequence of digits of pi?[/QUOTE]

[url]http://oeis.org/A060421[/url] supposedly shows that one known one is up to over 78000 digits depending on how you define a [URL="http://mathworld.wolfram.com/Pi-Prime.html"]pi prime[/URL]

[QUOTE=science_man_88;307603][URL]http://oeis.org/A060421[/URL] supposedly shows that one known one is up to over 78000 digits depending on how you define a [URL="http://mathworld.wolfram.com/Pi-Prime.html"]pi prime[/URL][/QUOTE]

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

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.

[QUOTE=Batalov;307923]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.[/QUOTE]

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

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

[QUOTE=Batalov;307923]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.[/QUOTE]

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.

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

[QUOTE=Batalov;308151]They are longer than 103,000 digits. :-)[/QUOTE]

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

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?

a(96) = 140,165-digit PRP
 

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 [strike]am DCing[/strike] 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:
[code]# 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
[/code]

a(20) is still ongoing.

EDIT2: strictly speaking, because a(96) is quite big - it [I]may[/I] not be a minimal solution: there's a chance that by way of some bug I [I]could[/I] 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.)