Primes in ϕ (phi)
 

Following the [URL="http://www.mersenneforum.org/showthread.php?t=16978&highlight=pixsieve"]Primes in π[/URL] thread, I've decided to start this thread. ϕ is also known as the golden ratio. The first million digits can be found [URL="https://www.goldennumber.net/phi-million-places/"]here.[/URL]

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 [URL="http://oeis.org/A064117"]this OEIS sequence[/URL] and [URL="http://oeis.org/A064119"]related sequence[/URL]. There is more information over at [URL="http://mathworld.wolfram.com/Phi-Prime.html"]MathWorld[/URL].

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

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.

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.

#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[/CODE]

#44 nothing up to 100K

[QUOTE=rogue;451068]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.[/QUOTE]

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.

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

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

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

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

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.

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.