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 davar55 2012-07-15 18:51

Primes in π

Finding primes in the digits of pi has been done,
so here is yet another primes in pi puzzle.

For every positive integer n (in decimal) find
the first occurrence in pi of the digits of that integer,
then the first prime constructed from the subsequent digits of pi.

Here's what I had in mind:
__________________________________
1 --> 14159
2 --> 2 [STRIKE]26535897932384626433[/STRIKE] (*)
3 --> 3
4 --> 41
5 --> 59 (*)
6 --> 653
7 --> 79 (*)
8 --> 89
9 --> 9265358[COLOR=sienna]97[/COLOR]9323 [COLOR=darkred](or 97 ??)[/COLOR]
10 -> [COLOR=#8b0000]102701 [COLOR=black](or[/COLOR] [/COLOR][COLOR=darkred]1058209749...6531873029[/COLOR][SUB]<19128>[/SUB] ?) (PRP)
11 --> 11
12 --> 12847564823 (or[COLOR=#8b0000] 12848111...678925903[/COLOR][SUB]<211>[/SUB] ?)
...
[COLOR=green]20 --> [COLOR=darkred](...more than 215000-digit...)[/COLOR][/COLOR]
[COLOR=green][COLOR=darkred]...[/COLOR]
62 --> 3490-digit Prime (and three more PRPs)
80 --> 41938-digit PRP.
81 --> 4834-digit PRP.
84 --> 3057-digit PRP.
96 --> [B][COLOR=darkred]140165-digit PRP[/COLOR][/B].
98 --> 61303-digit PRP.[/COLOR]
[COLOR=black]up to 100 (except 20): all primes/PRPs are less than 1000-digits or shown above[/COLOR]
__________________________________

(*) of course 2, 5 and 7 are prime
(**) My calculator only checks for small factors, so * and ** may not actually be prime

I think the list carried to (at least) 100 will possibly contain some more interestingly large primes,
or will the prime-checking facility be tested only by going to 1000, or beyond?
_______________

[COLOR=darkred]P.S. I could restore the original values for 2 and 10, but ... they were composite. You can easily see the original in post #2 (SB)[/COLOR]

 xilman 2012-07-15 19:07

[QUOTE=davar55;304822]Finding primes in the digits of pi has been done,
so here is yet another primes in pi puzzle.

For every positive integer n (in decimal) find
the first occurrence in pi of the digits of that integer,
then the first prime constructed from the subsequent digits of pi.

Here's what I had in mind:

1 --> 14159
2 --> 26535897932384626433 (*)
3 --> 3
4 --> 41
5 --> 59 (*)
6 --> 653
7 --> 79 (*)
8 --> 89
9 --> 9265358979323
10 -> 1058209749445923078164062862089986280348253421170679821480865132823 (**)
11 --> 11
12 --> 12847564823

(*) of course 2, 5 and 7 are prime
(**) My calculator only checks for small factors, so * and ** may not actually be prime

I think the list carried to (at least) 100 will possibly contain some more interestingly large primes, or will the prime-checking facility be tested only by going to 1000, or beyond?[/QUOTE]Another, related problem, would be to to find the first radix-n representation of primes in the radix-n representation of \pi which begin with the digit n-1 for all integer n >=2.

Alternatively, those primes which begin with each of the digits <n in the radix-n representation of the primes and of \pi

Paul

 patrik 2012-07-15 20:16

26535897932384626433 = 150917801 x 175830139033
1058209749445923078164062862089986280348253421170679821480865132823 = 196699 x 1834313671 x 4817619413830406641955201 x 608784400187359263779612387

 Batalov 2012-07-15 20:48

For 12, the value appears to be
[CODE]1284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903[/CODE]

But for 10, ...that's a real beauty!
Makes one wonder that the first possible chain of digits might go without a prime (can it? on a simple probabilistic argument?), then the next one easily produces [SPOILER]102701[/SPOILER] (*** same below)

13 --> 13
14 --> 14159
15 --> 1592653
16 --> 1693
17 --> 17
18 --> 1861
19 --> 19
20 --> 20................................... or 2089 (***)
21 --> 211
22 --> 223

 Batalov 2012-07-15 23:47

There shouldn't be a probabilistic argument, really. (Silly me.)

On one hand, these candidates are not sparser than any quasi/near-repunits (one per a power of 10 - except for some series with algebraic compositeness), -- and we have tons of primes/PRPs for them.

On the other hand, I actually found the 19128-digit PRP (easy to recontruct from Pi, code is below; decimal view: 1058209749...6531873029). Now we need a healthy volunteer to prove it; it would be a Primo record!

[CODE]# gp
\p 20000
write("p10",floor(Pi*10^19176)%(10^19128))
\q
pfgw -f -tc p10
[/CODE]

I still don't have a PRP for 20.............. though.

 davar55 2012-07-16 03:35

I appreciate the editing of my OP.
It was after all just a starting point.
For the sake of completeness, since 10..... and 20.....
are producing long sequences, it might be interesting
to check 30....., 40....., etc.

 Batalov 2012-07-16 04:00

I hoped that you wouldn't mind. Thank you for a nice problem!

30 and 40 turned out to be easy. 20 is still running empty (at least 9300 digits in it).

In the meantime, I installed a shiny new Bosch dishwasher to please SWMBO. "Change of work is rest", they say? :-)

EDIT: ...oh and for 2, I found a 50-digit prime, but then again, the mere "2" is already prime.

 bearnol 2012-07-16 12:44

You're aware that the first prime-instance might not necessarily start at the first instance?
(this is the same mistake as Shallit made)
J

 henryzz 2012-07-16 13:59

1 Attachment(s)
The following pari code finds solutions for 2 digit starting points upto length 1000.
[CODE]\p 20000
{
found=0;
for (n=10, 99,
for (offset=0, 1000,
if ((floor(Pi*10^(2+offset-1))%(10^2))==n,
for (digits=2, 1000,
f=factor(floor(Pi*10^(digits+offset-1))%(10^digits),9);
if (matsize(f)==[1,2],
if (ispseudoprime(floor(Pi*10^(digits+offset-1))%(10^digits),20),
print(floor(Pi*10^(digits+offset-1))%(10^digits));
found=1;
break;
);
);
);
if (found==0,
print("A solution has not been found for " n);
);
found=0;
break;
);
);
);
}
\q
[/CODE]

I have attached solutions for 10-99 that I have found. The following have no solutions upto 1000.
[CODE]10
20
62
80
81
84
96
98[/CODE]

All this is ignoring what bearnol just pointed out.
I will now write a script that produces input to pfgw for the harder numbers.

Is there a way of redirecting the output from a pari script without getting things like the header as well? I am a bit of a pari novice.

 xilman 2012-07-16 15:55

[QUOTE=henryzz;304913]Is there a way of redirecting the output from a pari script without getting things like the header as well? I am a bit of a pari novice.[/QUOTE]Yes.

Here's how the Perl script which is used to update my factor table tests its argument for primality.

[code]

# Primality testing function.

# Initial sanity check to see whether Pari/gp is installed and working correctly.

my $sc1 = echo "isprime(1074884750872101952308847649628260864479,2)" | /usr/bin/gp -f -q; # Known prime. my$sc2 = echo "isprime(1074884750872101952308847649628260864481,2)" | /usr/bin/gp -f -q; # Known composite.
($sc1 != 1 or$sc2 != 0) and die "Failed gp sanity check\n";
sub is_prime($) { my$num = shift;
my $big_mem = length$num > 300 ? 'allocatemem(104857600);' : '';
return echo "${big_mem}isprime($num,2)" | /usr/bin/gp -f -q  == 1;
}
[/code]

You should be able to read Perl well enough to translate that function into the language of your choice.

Paul

Paul

 Batalov 2012-07-16 18:43

[QUOTE=bearnol;304883]You're aware that the first prime-instance might not necessarily start at the first instance?
(this is the same mistake as Shallit made)
J[/QUOTE]
Would you care to expand on this?
How would you move on from the first instance to the next one?
By proof?
For example could you prove that there can not be a prime in these series: 1) 7019*10^n-1 or in 2) 8579*10^n-1. (subsequences of pi would be obviously harder)

[SPOILER]Yes, 2) is a trick proposition. There exists a prime.[/SPOILER]

 Batalov 2012-07-16 20:54

For 62, the PRP is 3490-digit.
For 81, the PRP is 4834-digit.
For 84, the PRP is 3057-digit.

(these would be easy to prove prime)

 henryzz 2012-07-16 21:22

[QUOTE=Batalov;304950]For 62, the PRP is 3490-digit.
For 81, the PRP is 4834-digit.
For 84, the PRP is 3057-digit.

(these would be easy to prove prime)[/QUOTE]

How far are you testing?

 Batalov 2012-07-16 21:25

I have 50000 Pi digits dumped from gp and then prefiltered by a simple perl script that takes care of small factors of 2,5 (easy!) and 3 (simple sum of digits). 7 and 11 could be easily added but is not of significant help.

Then the candidate file goes to pfgw -f. All very easy, very 'umble.

For 20, 80, 96 and 98, the PRPs would be larger than 30,000 digits now.

 Random Poster 2012-07-17 07:17

[QUOTE=Batalov;304937]Would you care to expand on this?
How would you move on from the first instance to the next one?
By proof?[/QUOTE]
By using a not-completely-insane method. In the outer loop, you take the first N digits of pi; in the inner loop, you take the last M digits of those N digits and test if that number begins with the specified digit(s) and is prime. Optimizations are possible: the outer loop can skip those N where the last digit is even, and the inner loop can keep a table of the positions of the starting digit(s).

 Batalov 2012-07-17 07:57

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).
[CODE]pi = 3.
14159265358979323846264338327950288419716939937510
58[COLOR=darkred][B]20[/B]974944592307[/COLOR]816406286[COLOR=blue][B]20[/B]8998628034[/COLOR]8253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196...
[/CODE]
Suppose we've checked the substrings starting from the red [COLOR=darkred][B]20[/B][/COLOR] 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 [COLOR=blue][B]20[/B][/COLOR], find [COLOR=blue][B]20[/B]89[/COLOR] and say that we are done.
We can only move on to the next instance of the starting point [I]after[/I] we have proven that there are no primes formed by the first instance. Can we prove that?

 axn 2012-07-17 08:21

Do you (the general "you") define the "first prime instance" using the starting position or the ending position?

Explicit clarification is called for.

 Batalov 2012-07-17 08:29

The problem is stated somewhat loosely and is open to interpretations.
Clearly we can all agree that the first, say, '3' after the prefix is not what OP intended. (Even though he wrote, "the first prime constructed from the subsequent digits".)

But I can see how you can interpret that as the string becomes longer it absorbes more instances of the prefix and you can hop onto that prefix. I think that this variant is too easy. It probably can be solved up to 5-6-digit values. Too easy.

Indeed, I am interested in the hard variant (For a given number [I]n[/I], find the leftmost instance of a prime substring in decimal expansion of pi that starts with [I]n[/I].) It can still be not the first useable prefix.

P.S. For the easy interpretation, 1 --> 41, isn't it? Maybe not (that's yet another interpretation).
But surely, for the easy interpretation, 9 --> 97

 bsquared 2012-07-17 15:14

[QUOTE=Batalov;305006]T
We can only move on to the next instance of the starting point [I]after[/I] we have proven that there are no primes formed by the first instance. Can we prove that?[/QUOTE]

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.

edit: here is where it would be nice for RDS to come around, slap me upside the head, and give a correct answer. I just put my $0.02 worth of thought into it.  Batalov 2012-07-17 18:26 80 -> 8034825342...6216977871[SUB]<41938>[/SUB] Formula: floor(Pi*10^42021)%(10^41938). (Submitted to Lifchitz&Lifchitz...) edit: I just put my$0.25 in the swear jar. B->

 davar55 2012-07-17 20:42

Indeed, the problem being solved by Batalov and henryzz et.al.
is the one I intended in the OP. Only if no prime exists in pi at
a found point should the next occurrence of the integer be used.

I see why this wasn't all clear in the OP.

 Dubslow 2012-07-18 03:00

I Am Not A National Treasure???

 bsquared 2012-07-18 04:16

[QUOTE=Dubslow;305072]I Am Not A National Treasure???[/QUOTE]

Clearly not :) but also not a Number Theorist.

 Batalov 2012-07-18 18:54

98 -> 9862803482...07182848167[SUB]<61303>[/SUB] PRP

 kar_bon 2012-07-19 15:02

[QUOTE=Batalov;304950]For 62, the PRP is 3490-digit.
(...)
(these would be easy to prove prime)[/QUOTE]

... is proven prime [url=http://factordb.com/index.php?id=1100000000524129946]here[/url].

 Batalov 2012-07-19 17:01

a(20) and a(96) both would be larger than 71000 digits. Running up to 100k digits.

 kar_bon 2012-07-20 00:38

1 Attachment(s)
Here's some code for finding possible numbers of PI-digit-primes for testing with pfgw.

All needed info. are given in the attachment.

 Batalov 2012-07-20 01:13

Ah. Interesting to compare different programming styles.
Here's my scriptus.
[CODE]#!/usr/bin/perl -w
$N=(shift || '20'); # Pi is prepared by gp :: \p 100000; write("Pi",Pi) open IN, "Pi";$_=<IN>;
s/\s+$//;$l=length($_); for($i=0;$i<length($_) && (substr($_,$i,length($N)) ne$N);$i++) {} die unless substr($_,$i,length($N)) eq $N;$s3=substr($_,$i,1); # sum of digits for divisibilty-by-3 test
for($j=1;$j<$l-$i;$j++) {$s3+=substr($_,$i+$j,1); print substr($_,$i,$j+1),"\n" if(substr($_,$i+$j,1) =~ /[1379]/ &&$s3%3!=0);
}
#then run pfgw -f cfile
[/CODE]

 gd_barnes 2012-07-23 06:47

IMHO, it makes some of the sequences "uninteresting" if we allow the number itself as a prime. To make them more interesting, I think that only primes with digits added should be allowed. Doing this, we have the following smallest primes from the 1st post of this thread:

[code]
1 --> 14159
2 --> 26535897932384626433832795028841971693993751058209
3 --> 31
4 --> 41
5 --> 59
6 --> 653
7 --> 79
8 --> 89
9 --> 9265358979323
10 -> (41938-digit PRP already posted)
11 --> 1170679
12 --> 1284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903
[/code]

Using the restriction of disallowing the number (sequence) itself as a prime, does this affect any already calculated results for sequences > 12 ?

 kar_bon 2012-07-23 07:04

[QUOTE=gd_barnes;305563]Using the restriction of disallowing the number (sequence) itself as a prime, does this affect any results for sequences > 12 ?[/QUOTE]

13, 17, 19, 23, 29,... and many others (see file in post #9).

 gd_barnes 2012-07-23 07:08

[QUOTE=kar_bon;305565]13, 17, 19, 23, 29,... and many others (see file in post #9).[/QUOTE]

Ah very good. Based on that, I would pose it as an additional difficulty to the problem to find primes with digits added to the 2-digit prime sequences.

 Batalov 2012-07-23 07:16

17 gets in a spot of trouble [SPOILER]but it has a 6918-digit PRP[/SPOILER]. Others (I checked only a few ...up to 100... 200) escape easily.

 davar55 2012-08-10 01:27

Based on the OP looking for certain primes among the digits of pi,
where is the first occurrence of each successive prime in pi,
i.e. the first "2", ... , the first "97", etc. up to say 100000.
Indexing could begin with the 3 as 1 or 0.

There are repetitions and the sequence is not in numerical order.
(I have not computed this sequence.)

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

 kar_bon 2012-08-10 11:18

[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].

 Xyzzy 2012-08-10 14:52

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

 science_man_88 2012-08-10 21:36

[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]

 davar55 2012-08-14 14:32

[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.

 Batalov 2012-08-14 17:01

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 2012-08-14 19:27

[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.)

 davar55 2012-08-16 13:31

[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.

 Batalov 2012-08-16 16:26

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

 davar55 2012-08-17 00:26

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

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

 ixfd64 2012-08-17 04:34

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 2012-08-21 16:52

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.)

 LaurV 2012-08-22 05:17

Using a bit different logic I confirm all the PRP values with <200 digits found up to now. Moreover, if we let apart the leading "3" and use only the digits in the fractional decimal expansion, that would modify the primes for 3 and 31:

[CODE]
(11:46:26) gp > get_primes_in_pi(0,100,1,1)
Found 0 at position 32. Checking for prime ... Found: prp=2
Found 1 at position 1. Checking for prime ... Found: prp=14159
Found 2 at position 6. Checking for prime ... Found: prp=26535897932384626433832795028841971693993751058209
[COLOR=Red]Found 3 at position 9. Checking for prime ... Found: prp=35897[/COLOR]
Found 4 at position 2. Checking for prime ... Found: prp=41
Found 5 at position 4. Checking for prime ... Found: prp=59
Found 6 at position 7. Checking for prime ... Found: prp=653
Found 7 at position 13. Checking for prime ... Found: prp=79
Found 8 at position 11. Checking for prime ... Found: prp=89
Found 9 at position 5. Checking for prime ... Found: prp=9265358979323[/CODE]

[CODE]Found 30 at position 64. Checking for prime ... Found: prp=307
[COLOR=Red]Found 31 at position 137. Checking for prime ... Found: prp=317
[/COLOR]Found 32 at position 15. Checking for prime ... Found: prp=32384626433832795028841971693993751058209749445923078164062862089986280348253421
Found 33 at position 24. Checking for prime ... Found: prp=33832795028841971
Found 34 at position 86. Checking for prime ... Found: prp=348253
Found 35 at position 9. Checking for prime ... Found: prp=35897
Found 36 at position 285. Checking for prime ... Found: prp=3607[/CODE]

The last parameter is "only_non_trivial_primes" that is, extend the numbers if they are prime already, and the third parameter is "use only decimal expansion (ignore the leading 3)".

 Batalov 2012-08-22 05:54

For the leading zero, the following prime must be in octal*! :-)
(this doesn't change the answer though, it's still "02")

Also, I've revisited the larger PRPs and let the searches run for a while more and found a few more PRPs starting with the leftmost "62": 3490-, 7734-, 11111-, and 17155-digit (the last two are reportable to Lifchitz[SUP]2[/SUP])

______
*C convention. printf("%d\n", 052); will print 42

 LaurV 2012-08-22 09:07

[QUOTE=Batalov;308875]For the leading zero, the following prime must be in octal! :-)
[/QUOTE]
:razz:

Joking apart, I just did a re-check for all thingies under 10k digits. With this occasion I found out that everybody completely missed 97. It was prime by itself in the "trivial" case, so it was not mentioned in post #9, and it was forgotten after the rules changed. My pari found a [URL="http://factordb.com/index.php?id=1100000000530494297"]nice 821 digits[/URL] beauty for it starting from position 12.

 Batalov 2012-08-22 09:20

It was not forgotten in post #32. PRPs under 1000 digits are too easy to even mention. (And Lifchitz site has a cutoff of 10000 digits.)

Only 17 was slightly more challenging.

 LaurV 2012-08-22 09:50

Ah, ok then.

I anyhow reported to FDB the PRPs for 54 and 73 (with 499 respective 446 digits) which were not reported, after I re-discovered them, together with the PRP for 97 in discussion.

 davar55 2012-08-22 20:03

Were you doing a(20) and a(96) in parallel?
So is length of a(20) already known > length of a(96),
assuming it resolves finitely?
Great work.

 zhongbii 2012-08-23 06:59

You can look in another way : Is the first N digit of pi (including 3)is a prime ?

Have a look at this
3
31
314159
31415926535897932384626433832795028841

what is the next "PI-PRIME"?

ps I'm poor in English ..... sorry

 Batalov 2012-08-23 18:51

Yes, this is the sequence [URL="http://oeis.org/A005042"]A005042[/URL]
(the extended version of the [URL="http://oeis.org/A060421"]A060421[/URL] sequence). We've already discussed these above. I suspect that multiple people searched for larger members of this sequence (in other words, we shouldn't think that the search stopped at the 78073; E.W.W.'s [URL="http://mathworld.wolfram.com/Pi-Prime.html"]mention[/URL] of the upper search limit is 6 years old).

 davar55 2012-08-31 14:04

The OP defined a single sequence, but somewhat loosely.
There are really an infinite number of sequences f[sub]i[/sub]
with the OP defining f[sub]1[/sub].
In that sequence, though it wasn't perfectly clear due to the
calculations presented, the primes were intended to be represented
by themselves (e.g. a(2) = 2 not the P50 that was found ).
But the examples showed that the OPer was uncertain about that point.

So f[sub]2[/sub] would be the sequence of primes starting at all the
same places in pi but the SECOND prime found. Similarly for f[sub]3[/sub]
and up.

I think just the first two sequences would cover all that the OP intended,
but finding the primes starting at ANY point in pi (as e.g. from the 3 prefix,
which is represented in the oeis) will lead to a somewhat interesting sequence.

 davar55 2012-09-21 18:35

Considering the surprising (to me at least) length of some of the a(*) being
discovered just up to 100, especially at 10, 20, 96, and 98, I think this sequence
is interesting enough to beg another question: Just how random are the digitis
of pi really? If we were to generate oher such "random" sequences (perhaps
the digits of e as transcendental or sqrt 2 as merely irrational but non-patterned),
seeing similar prime subsequence patterns might make this worthy of number theoretical
study.

In any case, as merely observor now, may I ask:
Is iit very hard to prove the biggest PRPs prime?
What's the L&L accreditor you referred to?
Is a(20) still chugging away?

Thanks for all your great work.

 CRGreathouse 2012-09-21 21:54

[QUOTE=Batalov;309040]E.W.W.'s [URL="http://mathworld.wolfram.com/Pi-Prime.html"]mention[/URL] of the upper search limit is 6 years old[/QUOTE]

He since increased it to 127,523 if I read this correctly:
[url]http://mathworld.wolfram.com/IntegerSequencePrimes.html[/url]

 davar55 2012-10-05 21:20

[QUOTE=CRGreathouse;312351]He since increased it to 127,523 if I read this correctly:
[URL]http://mathworld.wolfram.com/IntegerSequencePrimes.html[/URL][/QUOTE]

That appears to be the latest search limit, not itself prime; if I read that
list correctly. So a(96) is longer (yay). News on a(20)?

 Batalov 2012-10-05 21:53

No, ran some more than rebooted ...and into the bit bucket the trail went.

I was going to refactor (probably with a "real" sieve) and then rerun a(20) in another base from the start. The previous perl sieve was a toy:
[CODE]for($j=1;$j<$l-$i;$j++) {$d=substr($_,$i+$j,1);$s=(10*$s+$d) % 999999;
print substr($_,$i,$j+1),"\n" if($d =~ /[1379]/ && gcd($s, 999999)==1); #this takes care of 2,3,5,7,11,13,37 } [/CODE]  davar55 2012-10-06 00:44 [QUOTE=Batalov;313775]No, ran some more than rebooted ...and into the bit bucket the trail went. I was going to refactor (probably with a "real" sieve) and then rerun a(20) in another base from the start. The previous perl sieve was a toy: [CODE]for($j=1;$j<$l-$i;$j++) {
$d=substr($_,$i+$j,1);
$s=(10*$s+$d) % 999999; print substr($_,$i,$j+1),"\n" if($d =~ /[1379]/ && gcd($s, 999999)==1);
#this takes care of 2,3,5,7,11,13,37
}
[/CODE][/QUOTE]

Are all these perl variables automatically strings?
W/O any declarations, and not yet having used perl,
I wasn't sure. Looks like they're pre-initialized - to null?
(I'm referring particularly to $l and$i in the for loop control,
but they can't both be null - oh well).
I see it's basically C. I'll have to look up perl.

 Batalov 2012-11-05 23:19

1 Attachment(s)
This is just a replacement shippet of the full code (which was posted earlier), where $p is a long string with 250000 digits of [TEX]\pi[/TEX]following "20" ($i is the offset of "20"). I've attached here the latest variation, that saves on disk space and spawns pfgw on chunks.

[QUOTE=Batalov;308790]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). a(20) is still ongoing.

EDIT2: strictly speaking, because a(96) is quite big - it [I]may[/I] not be a minimal solution: ...[/QUOTE]
a(96) is double-checked (this time using base 2) and is the minimal solution. The gaps, if there were any, are now covered.

a(20)>10^215000. I've stopped the search.

 davar55 2012-11-28 20:20

Does anyone think this series is oeis-worthy?
As OP I'm not entitled to an opinion.
But some of those PRPs batalov discovered
are eye-worthy, I think.
Maybe when a(20) is finally tackled ...

 davar55 2013-08-16 11:13

[QUOTE]
a(96) is double-checked (this time using base 2) and is the minimal solution. The gaps, if there were any, are now covered.

a(20)>10^215000. I've stopped the search.[/QUOTE]Any news? Have you run it further?

What would it take to PRP test an unknown (C or P)215000?

Can you tell us the first 200 digits in pi of a(20)?

 kar_bon 2013-08-16 13:06

Generate your Pi with [url=http://www.numberworld.org/digits/Pi/]y-cruncher[/url].

 davar55 2013-08-16 13:15

Wouldn't want to crunch too many y''s, they're rare enough.

I see the first 20 occurs at digit 53.

Thanks.

 danaj 2013-08-23 15:41

1 Attachment(s)
I decided to give this a go, since it lends itself well to Perl, and seemed like a useful test case for my module. Attached is a simple serial script that outputs the result (using the "skip uninteresting" criteria) with a BPSW test all done in Perl, and no need for presieving etc. Thanks to Batalov for the initial script.

I've gone through all the numbers to 100 other than a(20) which I've got to 231k. All results pass BPSW, M-R with bases 3 and 5, and the Frobenius-Underwood test. I also had it run a proof (BLS75 theorem 5 or ECPP) on anything under 1000 digits. All of this is easily available in Perl. I used the "skip uninteresting numbers" variant, and I start with the first occurrence of the target number, so a(10) = PRP(19128), a(12) = PRP(211).

I also wrote a threads version where multiple threads pull lengths off a shared queue. I ran it for a(20) for a while on 48 cores of a Power7 machine. Starting at 214000 it got to 231000 before I had to quit. I'll run it with 12 threads on my Intel machine next. It's ugly managing the locks, writing out checkpoint marks, and making sure we output the minimum digits found rather than the first one found. I can post a link if anyone wants it.

To get the 'Pi-2M.txt' file I used y-cruncher, but at only 2M digits there are plenty of other tools that would do it.

The more interesting numbers, with their PRP digits:
10 -> 19128
17 -> 6918
20 -> ??? >231000
62 -> 3490
80 -> 41938
81 -> 4834
84 -> 3057
96 -> 140165
97 -> 821
98 -> 61303

[COLOR=DarkOrchid][Digressions ahead][/COLOR]

This is convenient, but is it fast? The BPSW test is faster than Pari, and the next_prime function is much faster than PFGW 3.7.7 for 2000+ digit inputs (digressing even more here, but perhaps PFGW isn't doing any sieving of candidates, or perhaps I'm using it wrong. "say next_prime(2**6600)-2**6600" takes ~30s for my code, "nextprime(2^6600)-2^6600" takes a bit under 2 minutes for Pari 2.6.1, while "./pfgw64s -V -q'nextprime(2^6600)'" takes over 4 minutes). However, PFGW's PRP tests are definitely faster (3.7.7 AVX FFT).

Well, everything is looking fine until we get into 30k+ digits. PFGW is much faster there (I imagine those familiar with it are eminently unsurprised). There isn't much I can do about the PRP tests, but it did give me the push I needed to put in a rudimentary treesieve (thanks to Jens Andersen for the nice writeup), which speeds up my trial division step for big numbers. Probably the fastest complete solution would be to put in a function like is_smooth($n,$B) that would just do the initial trial division, then we let PFGW handle the PRP tests on what's left.

To give some idea of just how bad this gets, for a(96), a 140,165 digit PRP on my computer:

[FONT=Courier New] 5m 50s PFGW (Fermat)
6m 15s PFGW (F-strong)
26m 30s PFGW (Fermat and [COLOR=DarkSlateGray]Lucas[/COLOR] PRP)
30m 15s MPU::GMP is_strong_pseudoprime($n,2) (1 [COLOR=DarkGreen]M-R[/COLOR]) 63m 40s Pari 2.6.1 ispseudoprime($n,1) (1 [COLOR=DarkGreen]M-R[/COLOR])
85m 45s MPU::GMP ES [COLOR=DarkSlateGray]Lucas[/COLOR] test
90m 15s MPU::GMP Frobenius-Underwood test
115m 55s MPU::GMP is_prob_prime (ES [COLOR=Navy]BPSW[/COLOR])
127m 0s MPU::GMP strong [COLOR=DarkSlateGray]Lucas[/COLOR] test
206m 20s mpz_prp is_strongbpsw_prime (S [COLOR=Navy]BPSW[/COLOR])
213m 15s Pari 2.6.2 ispseudoprime(\$n) (AES [COLOR=Navy]BPSW[/COLOR])
[/FONT]
It's much closer at 10k digits. Also until we hit the PRP at this range basically everything would be found composite by the SPSP-2 test, so we just pay the cost of the SPSP test until we found the prime where we do the full BPSW cost. The result has also passed BPSW when we're done, which we probably would have wanted to do anyway.

 J F 2013-09-02 23:04

1 Attachment(s)
Hi!

I did read this (then long dead) thread in early summer
and played around. Now some activity again, maybe
someone is interested in some newer numbers.
It seems me setting the end point to 1111 was
a bit optimistic... :blush:

 danaj 2013-09-04 00:04

JF, very impressive. What did you use for the primality testing?

I noticed you're not including the initial 3 in your search (e.g. you have 314 starting at 2120 instead of 0). I guess it's a bit ambiguous, but I read "digits of Pi" rather than "decimal digits of Pi".

I suspect dropping the "uninteresting" restriction I used (where we don't allow the result to be equal to the number) is probably best if the series is extended far enough.

There are a couple typos in the results:
- number 119 the start is at 494, not 404
- number 471 the length is 8610, not 8619

With corrections above, all results from your list with length < 100,000 verified with BPSW.

 LaurV 2013-09-04 02:45

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 [URL="http://www.mersenneforum.org/showpost.php?p=308871&postcount=45"]post #45[/URL], onwards).

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

Beside of it, the 10, [URL="http://factorization.ath.cx/index.php?id=1100000000530947423"] 17,[/URL] 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 2013-09-04 05:18

[QUOTE=LaurV;351795]Nice work. However, extending the sequence to higher starters is not big deal, as the most of them are "easy".[/QUOTE]But there are 8 more in his results that are indicated at 170k+ digits or more, so not all are easy.

[quote]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.[/quote]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.

[quote]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 [URL="http://www.mersenneforum.org/showpost.php?p=308871&postcount=45"]post #45[/URL], onwards).[/quote]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 [[I]positive[/I]] number of subsequent digits. Without "positive" we would accept 0 additional digits, otherwise not. The "uninteresting" bit came from [URL="http://mersenneforum.org/showpost.php?p=305563&postcount=29"]post 29[/URL] 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.

[quote]The only interesting case remaining after Batalov's work is actually "20". Who can solve the 20 gets a bonus... :smile:[/quote]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 2013-09-04 05:43

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

[COLOR=White](that was a compliment, told with much respect. don't get fussy about it!)
[/COLOR]

 danaj 2013-09-04 05:57

:( Sorry.

 J F 2013-09-04 10:03

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 2013-09-04 11:04

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 2013-09-04 16:11

"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. [URL="http://mathworld.wolfram.com/Pi-Prime.html"]Pi-Prime[/URL] (OEIS [URL="http://oeis.org/A005042"]A005042[/URL]) 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 2013-11-06 18:59

1 Attachment(s)
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 2013-11-08 00:40

[QUOTE=bsquared;305025]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.[/QUOTE]

This would also appear to follow if the digits are [url=http://en.wikipedia.org/wiki/Normal_number]normal[/url], 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 2014-08-17 11:28

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

 J F 2014-09-23 20:10

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

 davar55 2014-09-24 19:17

[QUOTE=J F;383738]a(20) neares 450k digits, still nothing
a(196) finished with a 312306-digit PRP[/QUOTE]

Re: a(20) at 450K: what's the probability at this point that
it will reach or exceed 1M digits ? If so, how expensive will
it be to check that far ?

 J F 2014-09-24 21:18

[QUOTE=davar55;383775]Re: a(20) at 450K: what's the probability at this point that
it will reach or exceed 1M digits ? If so, how expensive will
it be to check that far ?[/QUOTE]
Quick and dirty calculation, chance for (at least) one PRP from
450K to 500K 4-5%, 500K to 1M ballpark 25%.
That doesn't sound too bad, but the required core-hours ARE bad!
With PFGW a PRP-test at 450K digits takes ~2.5h on an older
non-AVX I5-750 core, 900K digits would be ~10h, a modern AVX-core
is a bit less than half that time.
Many thousand of them after TF (+TF time) - there you go.

 davar55 2014-10-01 19:35

Thanks. Will this computation be continuing indefinitely, or is there
some software limit, or a time budget? I think we'll need a(20) to
have a satisfactory entry in oeis, if that's deemed ok.

 davar55 2014-11-29 14:56

[QUOTE=ewmayer;358683]This would also appear to follow if the digits are [URL="http://en.wikipedia.org/wiki/Normal_number"]normal[/URL], 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.[/QUOTE]

For pi, e and sqrt(2), is there any evidence beyond empirical observation of the so-far known digits that addresses
their normality, i.e. other than statistics? Are there any positive results toward this? I too "generally believe" they
are normal, but looking at the first say trillion digits of sqrt(2) is still subject to the law of small numbers considering
the infinity of its non-repetitiveness.

 davar55 2014-12-14 15:59

I was very excited when I thought of the OP question. In attacking the
question by hand and with my calculator, I anticipated some very long
primes coming out of the digits of pi. Now I eagerly await a(20).

 davar55 2014-12-14 18:50

In preparation for a possible oeis entry, and since the PRPs here were not posted,
could someone post a list of the first 99 (01 thru 99) elements of this sequence?

Each line as:

xxx -> #

where xxx is the index (001 thru 099) and
# is either the prime, PRP### followed by abcde.....vwxyz (the end digits), or "no such prime".

(If you want to improve on this format, fine.)

 J F 2014-12-17 11:07

[QUOTE=davar55;390038]...and since the PRPs here were not posted...[/QUOTE]
??
Some outdated limits for the unfinished numbers and finished
#196 aside, what's wrong with the list in post 74?

Completely off topic: can someone recommend a program or
efficient way to cut down an arbitrary integer to a shorter
representation? Let's say I have a 1M digit number and want
it down to a 30? 20? chars term of form a*b^c+d.
As short as possible with 'reasonable' (minutes? no idea
about the complexity) CPU time.
Thanks in advance.

 fivemack 2014-12-17 11:31

Obviously it will nearly always not be possible to get such a representation, but I'd be tempted to use the linear-forms stuff in Pari on log(N) and logs of enough small numbers to try to fit A,B,C, then figure out D by subtraction.

[code]
lp=[];forprime(p=2,200,lp=concat(lp,[log(p)]))
lindep(concat([log(861*136^997+142857)],lp))
[/code]

312ms to recognise 85*53^2269-11 using primes up to 100 (with realprecision=500); 2152ms using primes up to 200; 5512ms for primes up to 300; you get a non-sparse vector if it didn't work.

 axn 2014-12-17 12:56

[QUOTE=J F;390274]Completely off topic: can someone recommend a program or
efficient way to cut down an arbitrary integer to a shorter
representation? Let's say I have a 1M digit number and want
it down to a 30? 20? chars term of form a*b^c+d.
As short as possible with 'reasonable' (minutes? no idea
about the complexity) CPU time.
Thanks in advance.[/QUOTE]

Simple information-theoretic reasoning suggests what you want cannot be done (i.e, compressing an [B]arbitrary[/B] integer). If it could be done, you've just found out the world's most efficient compression algorithm :shock:

 J F 2014-12-17 13:58

Shame - talk about obvious, I could have come up
with this myself...
I had some muddy thoughts similiar to Mr. Fivemack,
with X=a*b^c+d try to puzzle the logs together to
come close up to log(X) and the add/substract 'the rest'.
At this moment I just didn't think about that this is
trying to map a set 1:1 to a much smaller set.
*in the corner, with red ears*

Thanks for the replys

 davar55 2014-12-17 20:18

[QUOTE=J F;390274]??
Some outdated limits for the unfinished numbers and finished
#196 aside, what's wrong with the list in post 74?
...
Thanks in advance.[/QUOTE]

Your welcome (OP).

 davar55 2014-12-19 14:12

[QUOTE=davar55;304822]Finding primes in the digits of pi has been done,
so here is yet another primes in pi puzzle.

For every positive integer n (in decimal) find
the first occurrence in pi of the digits of that integer,
then the first prime constructed from the subsequent digits of pi.

Here's what I had in mind:
__________________________________
1 --> 14159
2 --> 2 [STRIKE]26535897932384626433[/STRIKE] (*)
3 --> 3
4 --> 41
5 --> 59 (*)
6 --> 653
7 --> 79 (*)
8 --> 89
9 --> 9265358[COLOR=sienna]97[/COLOR]9323 [COLOR=darkred](or 97 ??)[/COLOR]
10 -> [COLOR=#8b0000]102701 [COLOR=black](or[/COLOR] [/COLOR][COLOR=darkred]1058209749...6531873029[/COLOR][SUB]<19128>[/SUB] ?) (PRP)
11 --> 11
12 --> 12847564823 (or[COLOR=#8b0000] 12848111...678925903[/COLOR][SUB]<211>[/SUB] ?)
...
[COLOR=green]20 --> [COLOR=darkred](...more than 215000-digit...)[/COLOR][/COLOR]
[COLOR=green][COLOR=darkred]...[/COLOR]
62 --> 3490-digit Prime (and three more PRPs)
80 --> 41938-digit PRP.
81 --> 4834-digit PRP.
84 --> 3057-digit PRP.
96 --> [B][COLOR=darkred]140165-digit PRP[/COLOR][/B].
98 --> 61303-digit PRP.[/COLOR]
[COLOR=black]up to 100 (except 20): all primes/PRPs are less than 1000-digits or shown above[/COLOR]
__________________________________

(*) of course 2, 5 and 7 are prime
(**) My calculator only checks for small factors, so * and ** may not actually be prime

I think the list carried to (at least) 100 will possibly contain some more interestingly large primes,
or will the prime-checking facility be tested only by going to 1000, or beyond?
_______________

[COLOR=darkred]P.S. I could restore the original values for 2 and 10, but ... they were composite. You can easily see the original in post #2 (SB)[/COLOR][/QUOTE]

In the OP, I had to enter the digits of pi by hand into my calculator, and let it test for primality.
That procedure was error-prone, hence the wrong values for 2 (extended from the true value of 2) amd 10.
But I hurried to get my partial results out, because I knew the forum would do the calculations over.

I think this will be an important sequence and jumping off point - we'll see.

 MattcAnderson 2014-12-27 18:07

3.14159 2654
And that's all I know. The primes
There are 2 3 5 7. And there are more.

Thank you very much for starting this thread

Regards

Matt

 davar55 2014-12-28 15:10

I've just noticed:

the series as computed (but not as in post#74) can be used to reconstruct the
digits of pi, as the first occurrence of the positive integers in pi exhaust pi as
the sequence of integers progresses. there is overlap, as in the first 3 (intended
to be the lead 3) and the first 31 (also from 3.1) and the first 314 (which can be
reconstructed from the 314th prime in the series (which starts with 314... ).

hence we should probably index the lead 3 as index 0 of the digits of pi.
this loses no primes, they just appear later in the series.

also, when we confirm the first prp of a(20), this reconstruction of pi from the
output would allow a double check of both the digits of pi and all the intermediate primes.

this assumes: For any positive integer written in decimal it is possible to construct a
prime number by concatenating some finite sequence of decimal digits at its end.

which is certainly true.

 davar55 2015-03-06 23:32

It's been a while. Any new results ?

Has a(20) been resolved with a prime, or probable prime,
or has anyone tried proving that a(20) never produces a prime ?

Any other big results below a(1000) ?

 davar55 2015-03-27 10:10

Any resolution of a(20)?

How long does each PRP check take at 450K?

 davar55 2015-04-12 00:47

[QUOTE=davar55;398758]Any resolution of a(20)?
How long does each PRP check take at 450K?[/QUOTE]

Hate being a pest but I'm really into this result.
Can you say how high a(20) has been tested?

:geek:

 davar55 2015-04-28 20:11

[QUOTE=davar55;399890]Hate being a pest but I'm really into this result.
Can you say how high a(20) has been tested?
:geek:[/QUOTE]

Is anyone here still working on this one?

 science_man_88 2015-04-29 00:20

[QUOTE=davar55;401142]Is anyone here still working on this one?[/QUOTE]

first thing that comes to mind is pari and using say FaR or my attempt at parvecsearch ( see pari commands thread) but with alteration and depending on which gets used to look for all primes in a given range that start with that and searching for them in parallel or something like that.

 danaj 2015-04-29 15:42

[QUOTE=science_man_88;401178]first thing that comes to mind is pari and using say FaR or my attempt at parvecsearch ( see pari commands thread) but with alteration and depending on which gets used to look for all primes in a given range that start with that and searching for them in parallel or something like that.[/QUOTE]
See post 64 for some timings once the size gets large. Pari is 10 times slower than PFGW once the sizes get very large. One still needs to run something like BPSW on the result (e.g. Perl/ntheory, Pari, or WraithX mpz_prp), but that ought to be faster than using Pari for each value.

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