Primes in π

[davar55]

Thanks for turning PI on its head in the title of the thread.
I was going to ask, and this gives me the opportunity:

How is the same OP question answered for 1 / PI ?

In other words, if the sequence of digits used is those of
the reciprocal of PI, what is the sequence of prime numbers
formed from the digits beginning at the first occurrence of
each of the consecutive positive integers ?
(Using the suggested rule that a prime integer is not its
own representative but must be extended to another prime.)

Since PI * (1 / PI) = 1, and since both PI and 1 / PI are thought
to be normal transcendentals, I'm willing to go out on a limb
and guess the result will be as interesting as PI's. I'm
particularly interested in where that first really long sequence
(if one like a(20) shows up) begins.

[davar55]

Originally Posted by davar55

Thanks for turning PI on its head in the title of the thread.
I was going to ask, and this gives me the opportunity:

How is the same OP question answered for 1 / PI ?

In other words, if the sequence of digits used is those of
the reciprocal of PI, what is the sequence of prime numbers
formed from the digits beginning at the first occurrence of
each of the consecutive positive integers ?
(Using the suggested rule that a prime integer is not its
own representative but must be extended to another prime.)

Since PI * (1 / PI) = 1, and since both PI and 1 / PI are thought
to be normal transcendentals, I'm willing to go out on a limb
and guess the result will be as interesting as PI's. I'm
particularly interested in where that first really long sequence
(if one like a(20) shows up) begins.

By the way, is it merely a conjecture, or has anyone proven,
that the reciprocal of a normal transcendental is also a
normal transcendental?

[davar55]

I'd like to get this started.

Here are the first 550 digits of 1 / pi.
Notice they start with .31 (like pi's 3.1), but
we can start counting place value right from
the first digit.

31830988618379067153776752674502872406891929148091289749533468811779359526845307
01802276055325061719121456854535159160737858236922291573057559348214633996784584
79933874818155146155492793850615377434785792434795323386724780483447258023664760
22844539951143188092378017380534791224097882187387568817105744619989288680049734
46954789192217966461935661498123339729256093988973043757631495731339284820779917
48278697219967736198399924885751170342357716862235037534321093095073976019478920
7295186675361186049889932706106543135510064406495556327943320458934962

These are the first 20 "primes" in the sequence b(1)...b(20).
The number in () is the starting position for the appropriate prime.
Notice b(19) is the longest, at 306 (prp) digits.
Also notice b(1)=b(18).

1 (2) -> 183098861837
2 (26) -> 2674502872406891929148091289749
3 (1) -> 31
4 (29) -> 450287
5 (19) -> 53
6 (9) -> 61
7 (13) -> 79
8 (3) -> 83
9 (6) -> 98861837
10 (358) -> 105744619
11 (64) -> 11779
12 (50) -> 1289
13 (466) -> 13392848207
14 (45) -> 148091
15 (18) -> 15377
16 (137) -> 1607
17 (65) -> 17793595268453
18 (2) -> 183098861837
19 (41) -> 
 19291480912897495334688117793595268453070180227605532506171912145685453515916073
 78582369222915730575593482146339967845847993387481815514615549279385061537743478
 57924347953233867247804834472580236647602284453995114318809237801738053479122409
 788218738756881710574461998928868004973446954789192217966461935661 <prp306>
20 (393) -> 207799

If we push this up to b(100) or b(1000), we may find some more very large PRPs
in the decimal expansion of the reciprocal of pi.

[davar55]

Could someone at least double-check that I computed the first digits of 1 / pi correctly?

[Batalov]

They are correct.
Is there a way to calculate them incorrectly?! Ah, I forgot. The pocket calculator. Can't trust it, can we?

? \p 2000
   realprecision = 2003 significant digits (2000 digits displayed)
? 1./Pi
0.31830988618379067153776752674502872406891929148091289749533468811779359526845307018022760553250617191214568545351591607378582369222915730575593482146339967845847993387481815514615549279385061537743478579243479532338672478048344725802366476022844539951143188092378017380534791224097882187387568817105744619989288680049734469547891922179664619356614981233397292560939889730437576314957313392848207799174827869721996773619839992488575117034235771686223503753432109309507397601947892072951866753611860498899327061065431355100644064955563279433204589349623919633168121203360607199626782397499766557330887055951014003248135512877769914262176024439875229536275552947578126613609291595696352262485462813992155004900059551971417811380559357026305042003263549204184962321248112291240629296817849691838287042315081511240174305321360443431828151494916544519549257079975031065878162796354481871650959414665743808139995181531541569869407871796561743468512807337902332509141188665526253730005224543594230642251990087733589007525112167263423390519516256449883246668629021224707375712622727338433428413949392025850115667210623921718901967911343741990949302086324763103516167888595994199901050877513225889176661369210157058303028208097859770127763215523939861468207799915738378119618747554412375086445437860273251052247756077507776221362813530868165655705386685359911214158077212070547799249025199149855259404718819116860232965928237115542481150889891404357953958481898065458954043329920713063630708800768137974943538317752638193301392880955394137536731355620955959090070679151660376367737587553224962990611993116043816719750207025425808646316099743937375551893132692442068408881710995700758547738858707323875565857471875686940646047429167584711423727268385892036636458392833001756615866270699558199491729858053490121978737818917661006740610761094624643161886395352064566262837961949964487667034871397969500207900136776007957344719921604800547802174990970957584713652227989780653799485416699222984165780755356948607101

[danaj]

I get the same result for 1/pi, but...

2 (26) -> 2674502872406891929148091289749

that number isn't prime. Nor is the number you show as a prp306.

Some of your starting points don't look right. 10, 13, 16 all have earlier positions.

[davar55]

Originally Posted by danaj

I get the same result for 1/pi, but...
2 (26) -> 2674502872406891929148091289749
that number isn't prime. Nor is the number you show as a prp306.
Some of your starting points don't look right. 10, 13, 16 all have earlier positions.

I'm finally learning to double-check my numbers. It's not my calculator that's at fault,
it's my procedures. When I eventually have time, I'll recalculate my OP.
But for now, would anyone post the corrected list corresponding to the 1/pi post?

[davar55]

Perhaps this puzzle needs a new thread.

[davar55]

The last result for a(20) mentioned in this thread,

a(19) = 197
a(20) = 2097494459... <unknown...450K+>
a(21) = 211

was tantalizing. Is there any further news?

[storflyt32]

Where is the P306?

Losing track of it?

[storflyt32]

http://factordb.com/index.php?id=1100000000822815966

Took a little time, but definitely worth it.