Primes in ϕ (phi)

rogue · 2017-01-15, 23:48

Following the Primes in π thread, I've decided to start this thread. ϕ is also known as the golden ratio. The first million digits can be found here.

There are two components to this search. The first is from the other thread:

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

Barring mistakes I have tested all starting values from 1 to 99. Only 15, 44, 70, 76, and 92 do not have primes under 1000 digits. Who wants to find a prime for those?

The second is to extend this OEIS sequence and related sequence. There is more information over at MathWorld.

J F · 2017-01-16, 09:33

#15 4050 digits, #44 nothing up to 15K.
(modulo me messing something up)
I will do some more later.

J F · 2017-01-16, 11:14

Now that was quick.
#70 7399 digits
#76 7045
#92 2692
#44 on its way to 100K, off to work,
lets see what we have tonight.

rogue · 2017-01-16, 20:26

Thanks.

I am commencing sieving on the first 1,000,000 digits of ϕ.

I originally started at 500,000, but after sieving to about 3e7 it was down to 17000 terms, so I expanded the range.

Antonio · 2017-01-17, 10:21

#15 - 4050 digits

Code:

158846074998871240076521705751797883416625624940758906970400028121042762177111777805315317141011704666599146697987317613
560067087480710131795236894275219484353056783002287856997829778347845878228911097625003026961561700250464338243776486102
838312683303724292675263116533924731671112115881863851331620384005222165791286675294654906811317159934323597349498509040
947621322298101726107059611645629909816290555208524790352406020172799747175342777592778625619432082750513121815628551222
480939471234145170223735805772786160086883829523045926478780178899219902707769038953219681986151437803149974110692608867
429622675756052317277752035361393621076738937645560606059216589466759551900400555908950229530942312482355212212415444006
470340565734797663972394949946584578873039623090375033993856210242369025138680414577995698122445747178034173126453220416
397232134044449487302315417676893752103068737880344170093954409627955898678723209512426893557309704509595684401755519881
921802064052905518934947592600734852282101088194644544222318891319294689622002301443770269923007803085261180754519288770
502109684249362713592518760777884665836150238913493333122310533923213624319263728910670503399282265263556209029798642472
759772565508615487543574826471814145127000602389016207773224499435308899909501680328112194320481964387675863314798571911
397815397807476150772211750826945863932045652098969855567814106968372884058746103378105444390943683583581381131168993855
576975484149144534150912954070050194775486163075422641729394680367319805861833918328599130396072014455950449779212076124
785645916160837059498786006970189409886400764436170933417270919143365013715766011480381430626238051432117348151005590134
561011800790506381421527093085880928757034505078081454588199063361298279814117453392731208092897279222132980642946878242
748740174505540677875708323731097591511776297844328474790817651809778726841611763250386121129143683437670235037111633072
586988325871033632223810980901211019899176841491751233134015273384383723450093478604979294599158220125810459823092552872
124137043614910205471855496118087642657651106054588147560443178479858453973128630162544876114852021706440411166076695059
775783257039511087823082710647893902111569103927683845386333321565829659773103436032322545743637204124406408882673758433
953679593123221343732099574988946995656473600729599983912881031974263125179714143201231127955189477817269141589117799195
648125580018455065632952859859100090862180297756378925999164994642819302229355234667475932695165421402109136301819472270
789012208728736170734864999815625547281137347987165695274890081443840532748378137824669174442296349147081570073525457070
897726754693438226195468615331209533579238014609273510210119190218360675097308957528957746814229543394385493155339630380
729169175846101460995055064803679304147236572039860073550760902317312501613204843583648177048481810991602442523271672190
189334596378608787528701739359303013359011237102391712659047026349402830766876743638651327106280323174069317334482343564
531850581353108549733350759966778712449058363675413289086240632456395357212524261170278028656043234942837301725574405837
278267996031739364013287627701243679831144643694767053127249241047167001382478312865650649343418039004101780533950587724
586655755229391582397084177298337282311525692609299594224000056062667867435792397245408481765197343626526894488855272027
477874733598353672776140759171205132693448375299164998093602461784426757277679001919190703805220461232482391326104327191
684512306023627893545432461769975753689041763650254785138246314658336383376023577899267298863216185839590363998183845827
644912459809370430555596137973432613483049494968681089535696348281781288625364608420339465381944194571426668237183949183
237090857485026656803989744066210536030640026081711266599541993687316094572288810920778822772036366844815325617284117690
979266665522384688311371852991921631905201568631222820715599876468423552059285371757807656050367731309751912239738872246
825805715974457404842987807352215984266766257807706201943040054255015831250301753409411719

J F · 2017-01-18, 14:16

#44 nothing up to 100K

rogue · 2017-01-18, 16:41

Quote:

Originally Posted by rogue

Thanks.

I am commencing sieving on the first 1,000,000 digits of ϕ.

I originally started at 500,000, but after sieving to about 3e7 it was down to 17000 terms, so I expanded the range.

Still sieving. I'm at about 3e9 and have 25,500 remaining terms out of the initial 1,000,000. The removal rate is about one every eight minutes.

J F · 2017-01-20, 10:13

#44 no PRP up to 125K. I now go back
to my pet with one letter less.

ericw · 2017-02-02, 15:23

FWIW, back on 2009-06-03, I used Mathematica to show

PrimeQ[FromDigits[First[RealDigits[GoldenRatio, 10, 97241]]]]
True

See http://mathworld.wolfram.com/IntegerSequencePrimes.html (under "phi-prime")

rogue · 2017-02-02, 16:20

Thanks. I'm at about 120,000 digits using a single core. I have sieved to 1,000,000 digits and there are a little over 23,000 tests in the entire range.

rogue · 2017-02-17, 17:37

I have tested to 200,000 digits and am continuing. Eric Weiss has previously only searched to about 190,000 so nothing new was expected up to that length.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Mersenne Primes p which are in a set of twin primes is finite?	carpetpool	Miscellaneous Math	3	2017-08-10 13:47
Distribution of Mersenne primes before and after couples of primes found	emily	Math	34	2017-07-16 18:44
Conjecture about Mersenne primes and non-primes v2	Mickey1	Miscellaneous Math	1	2013-05-30 12:32
A conjecture about Mersenne primes and non-primes	Unregistered	Information & Answers	0	2011-01-31 15:41
possible primes (real primes & poss.prime products)	troels munkner	Miscellaneous Math	4	2006-06-02 08:35

2017-01-15, 23:48	#1
rogue "Mark" Apr 2003 Between here and the 11·577 Posts	Primes in ϕ (phi) Following the Primes in π thread, I've decided to start this thread. ϕ is also known as the golden ratio. The first million digits can be found here. There are two components to this search. The first is from the other thread: For every positive integer n (in decimal) find the first occurrence in phi of the digits of that integer, then the first prime constructed from the subsequent digits of phi. Barring mistakes I have tested all starting values from 1 to 99. Only 15, 44, 70, 76, and 92 do not have primes under 1000 digits. Who wants to find a prime for those? The second is to extend this OEIS sequence and related sequence. There is more information over at MathWorld.

2017-01-16, 09:33	#2
J F Sep 2013 2³·7 Posts	#15 4050 digits, #44 nothing up to 15K. (modulo me messing something up) I will do some more later.

2017-01-16, 11:14	#3
J F Sep 2013 2³×7 Posts	Now that was quick. #70 7399 digits #76 7045 #92 2692 #44 on its way to 100K, off to work, lets see what we have tonight.

2017-01-16, 20:26	#4
rogue "Mark" Apr 2003 Between here and the 1100011001011₂ Posts	Thanks. I am commencing sieving on the first 1,000,000 digits of ϕ. I originally started at 500,000, but after sieving to about 3e7 it was down to 17000 terms, so I expanded the range. Last fiddled with by rogue on 2017-01-16 at 20:39

2017-01-18, 14:16	#6
J F Sep 2013 2³×7 Posts	#44 nothing up to 100K

2017-01-20, 10:13	#8
J F Sep 2013 2³×7 Posts	#44 no PRP up to 125K. I now go back to my pet with one letter less.

2017-02-02, 15:23	#9
ericw Oct 2015 19 Posts	FWIW, back on 2009-06-03, I used Mathematica to show PrimeQ[FromDigits[First[RealDigits[GoldenRatio, 10, 97241]]]] True See http://mathworld.wolfram.com/IntegerSequencePrimes.html (under "phi-prime")

2017-02-02, 16:20	#10
rogue "Mark" Apr 2003 Between here and the 11·577 Posts	Thanks. I'm at about 120,000 digits using a single core. I have sieved to 1,000,000 digits and there are a little over 23,000 tests in the entire range.

2017-02-17, 17:37	#11
rogue "Mark" Apr 2003 Between here and the 11×577 Posts	I have tested to 200,000 digits and am continuing. Eric Weiss has previously only searched to about 190,000 so nothing new was expected up to that length.