Thread for posting tiny primes - Page 42

CRGreathouse · 2010-09-09, 16:14

Quote:

Originally Posted by 3.14159

Why don't you do that?

No interest -- it's pure number-crunching. (The only reason this came up was my declination of this sort of brute force CPU work, remember?)

But since you seemed interested in that sort of calculation I thought I'd offer the chance. It's somewhat more lasting 'fame' than the Top 5000 list which has a good deal of churn.

3.14159 · 2010-09-09, 16:17

An example of a 2^a * 3^b + 1 prime:

2⁴³ * 3⁷⁹⁶ + 1 is prime.

Larger example: 2²⁰⁰ * 3⁴¹⁸³ + 1 is prime.

CRGreathouse · 2010-09-09, 16:26

Quote:

Originally Posted by 3.14159

An example of a 2^a * 3^b + 1 prime:

2⁴³ * 3⁷⁹⁶ + 1 is prime.

Erdős-Selfridge class 1- primes, Sloane's A005109. The biggest known, as far as I can tell, is 2⁶⁴⁸⁵⁵³ * 3²⁵⁶ + 1. (Edit: See below.)

3.14159 · 2010-09-09, 16:30

Actually, it is 3 * 2^5082306 + 1. (1529928 digits)

Never specified anything against 321 primes.

The largest 2 * 3ⁿ +1 prime is where n = 1175232, apparently, at about 560729 digits.

CRGreathouse · 2010-09-09, 16:42

Good call. My search was for

Code:

^[23]^%*[23]^%+1$

but I should have also checked

Code:

^[23]*[23]^%+1$

and

Code:

^[23]^%+1$

(not that more Fermat primes are likely...)

Of course technically the % is over-broad, but [[:digit:]]{1,} is so verbose...!

3.14159 · 2010-09-09, 16:51

Quick PRP: 17212* 3¹⁸⁸⁶⁰ + 1 (9003 digits)

Gained a spot for items 5 and 20:

Code:

17212*3^18860 + 1 may be prime. (a = 2)
17212*3^18860 + 1 is prime! (a = 5) [9003 digits]

I'll post the decimal expansion here:

Code:

552962080195460110567759587723652300261049730437483300121779312287569322854502429939330107423571434534829709195786019144253642385982949557384000210392178894662184296907769197923921000992532678577860269338230790818726641859166199289571340406818540899545013288617900473378394389686410091293280482125986137115184714581006544027076712065950916147169638721277925229128779732666729066598939550402258119717907349397914770414125197200892121435509770044104894801346403120591020092678242311254011896792234543191204304785326466418616797969701586012776741503059607965164082560013899521974600711045232871870579398575788942883851612069137773612153473057116991666268562470075758074755495320840998725812000286473508418775150586973734160799535668257946262862502919115814534651999431644568657865321433922542538909778536354617074682737998518556773020302871980636311960218567547351397602622076077407386351843980898453200177212060609268805029343618304667889837382384373196876909981758994302982909760374918995459620487098165411734898307804122882400362265593911569839331616992338721643007641246575112984521251027132526314930918494222903429441455699635299336495023021123257434220128286622985875556008359893892697910457070977018671382420348808017784773041664939230657380984034853210513517037849712660546139371383012552679255879584968811949682784145717942695834650759242663715680388784811748497868840771243085932635375308890401829144244362609725232447920181849469663324762726532093515452736371524589351292280109064196293033233291886812399235275194193466674610026827978176237252552420638154543135654652213025420962437542535392933068433875014127926977813829109008980920099908484224115684153659377316678917406069933846979442066423025608462090814256512129293255494027447335807157838934589357581740823910242050351464016855074035704541336753307696456934358867181381663855003260881922739418869493212325292712665155844890700424144431086008360171445030893581809553844230104107583624665531092596123033586126073400844787163231466690116678483015187376951556950680357335376408777046544718537419473666736250976391462190776181802143140220856356196357593114556981655181644250883022488006652491557970511295219922153579374431368921478068073617778895073208117772376564488654223019231943098480394090720144764443069864496551178950716680565420491842005613916344195606663206666249751503130993272825618304752203360471275398382495438952841253174291854486777411636718185465479993449114674790815761948015600229824834011626008444021298824612900162860861475734261373745948556424446311888378520479496783557300272256028154082052436411855070042906711092337703475679082989160076247724564309084475338190528344602701867006432503119065338300485041108266390448867116228522024779050571252142088366077963949669435028753893592044256274075901553869480471010338994497106095130303206305941260150780035030326216467992389566460896997837792485871465624809101171406758544362371314663658345399579611933448399358646416161920299909146245318365870874423584130424188305441367795522154352205375858537980201726357015566368795699809706576523345080184042471674415619893549687172525022594972454527010166781254155336608169943181121558801459280594841015658485030394891713072996874197359738007120649147457963770446000508710007289713678937527588143111810526818418684782902904933414391462135353535349744115135555536671054382998625208543358830762210883908961477659593175468557556457311129161108522492669619001104197330406809376353331267624366101727528931421175954568729509884043443182812887797603111300302636423652111494922552965487951934910176640611710505523551935427839946740679892742586159782698760164012216865789318082406882171376226144005617811151139644892995650407638366052822764167992636733151654811385955687181895470888515129746470921957380664906865760383489745564549030160906780533283188615702025057681175136950862176866083090529759496337673213206195138729081795725729327024121638541334291235166827546935540232281413296335422028798877782138796455353084534169700138590175420792304772348261121778897590249950920822834359668794980861284941442397120038615289732520211245108597238039241211554571562915212355135518925829055104972393882966277204175323124378465722504136110948458100633367477398010300807105661307627821248786080672313047124772354237739859942330159871945528292050734106337762420624172131095695637198855559747427019094748896175983786126498296458433080494669534716615237753960990392946118686942364229731866528451984921142199919327948098088812340295894998665905158530235256530618322614274964287326030574194903942526224663439020563451320826279378685967171478805853245100934780617729872563745399952412559394193012370492588927511497254083768772708170570010568207479037880675149177504083006969343240784906670461691688822178101717101695963735167264846588913382747128285888560424521552680414337748560189957065143000936929123007799690167665556669315690772914625022687988794205188024897369981532946307624190427077212165223675656222727248458020440634192432109614118078657032182452062472223429073575330751434013856927206414927854520717566388530279631898545578084801687905229916775828942257321740347534923242774461854784113350600775273575933209404367953580384963942281935115876727214212313180051691716403080322370328914661011616602619292474316058706122621095625670905619608670175009998756688358532573137962363328290651266396444758808133376324999663174443899740642785097975668551067947973602018659524783754986219308495334271044922946423600207910146363388546826319256110037899513478152850581351166879380257015462314890194905660536827463083105030881819559631000483357937012056189528581421736182927831069409621089202800472196101980420443529899917943092744828801486876366016172893150043769657686660801740411707444109480099625371191396060583443188574225029045756690415629357923462049339582829491050254715821738746687794106328392422617730717268148005332996053888589431137939081931853382210486001298673576705896863470833034728603166161087761443111027931250903994677841259865344103851692830341871817026502695232229135974073391190979565731531236575076227266778478577836291778508302688575775979962481110257558814941288753871115736711451759070220503483373223246744429102700907291108541109945078312331799530594142851984644035590617716694260806260701124138489366456395689220837406345852432645341899848380112930397438914958862814875996228054578763417396732531440483425217151240986532145353076875797556935480574525438761888410901223654656589972611907622009109713263336454988472303895115387860195452530392534776349723387673532312623886834191144453075998708625389520679310452548245739763043115535315253474261055004632634420503377716696951318723888225425792939689647517862292639129958972847571778823079219522779981994423405085229697196180124021910560936232404358448454276143008538558265686091593280509772907729862255358186653007505614157071418362031582383808967602007476687155529489539908388109167116626234863821193042266592596825065855321929219250662562138399660105609351051701677316338177833280268678064001604880430302303509768788178662078790230370688264307986500771276144042521173224763404011803971685682993229338331008611278910534372512419318549537213292562617806649648268187278731371457531406762193858118222836969238903155159409068415852058951299899179327142583673245586390326320084971394530782543923859299091933602460859335100215501390783418013560267842836219322304117175579951551298131403318439743283281134985383891416984308073493773921563816355535464005648713260369315429782553905443146697101422691073377493613597734224800983083523567539311714689066452659887404058139545507597832133567785249056943470725317642433160130020895826203961750137490239328343746019111447437842093222754741643957467074546613453221074220032003572011436337747442862711755037025318343044627295880495525480965802868480000043760622362817284999390243691160534561575146607255379305821346569725649693948834545509344120695090267365500142952441678576952360087271024926140619433670330875577854469634041657887880533018313910125040662150488808373829343410451844581101255484204924404740165320369713716088998517336252129436952448145334119358173317589017894152522212046662375124317480284505131845001063453236113377854858980226945063028092467075515949830873442787858137300662139106268885966917073401159957331942768276809991498410841225050572364856220357096428407748231487724998830971845496890122064563323986132565362101164929277675891561363281934588636835710512914276948109173334173105447107806318988910033365162440454479137835788611424500045147147165302708347929998439824718675502786340666910500685394292256039514454721708467396662298647902066250236313677743096364320783106014062088400876163205397359430368364978629091115774514328750715645169135747294539425052409859920547790897107608725280040064936297045561791616493272941524902560573294623690604015791846748880172623618231280966605759780286812882357760868842127112167576487482756379800867254454279658122264191872025903909636083123399373032931167634808629923420407496725923072861267739487613

3.14159 · 2010-09-09, 18:17

Next PRP: 27841 * 12¹¹²⁸⁰ + 1. (12178 digits)

Verification of primality:

Code:

Primality testing 27841*12^11280+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Special modular reduction using zero-padded FFT length 5K on 27841*12^11280+1
Calling Brillhart-Lehmer-Selfridge with factored part 55.77%
27841*12^11280+1 is prime! (4.0765s+0.0009s)

Another for item 2.

3.14159 · 2010-09-09, 22:46

The largest small prime (Less than 1000 digits) :

Code:

999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999993899

Followed by the smallest prime of 1000 digits:

Code:

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007

There are ≈ 4.349182827963454639308289810704221769 * 10⁹⁹⁶ small primes.

CRGreathouse · 2010-09-09, 23:53

See
http://oeis.org/classic/A074282
for other small gigantic primes. I don't know if there is a similar sequence on the other side.

Quote:

Originally Posted by 3.14159

There are ≈ 4.349182827963454639308289810704221769 * 10⁹⁹⁶ small primes.

Your value is off by a factor of 10. Also, don't show so many decimals -- most of what you give is wrong, even when the exponent is corrected.

kar_bon · 2010-09-10, 00:10

... and 10^999-6251 is the next smaller prime with 999 digits.

3.14159 · 2010-09-10, 01:21

Quote:

Originally Posted by Charles

Your value is off by a factor of 10. Also, don't show so many decimals -- most of what you give is wrong, even when the exponent is corrected.

As if what you give is any more correct.

2010-09-09, 16:17	#453
3.14159 May 2010 Prime hunting commission. 1680₁₀ Posts	An example of a 2^a * 3^b + 1 prime: 2⁴³ * 3⁷⁹⁶ + 1 is prime. Larger example: 2²⁰⁰ * 3⁴¹⁸³ + 1 is prime. Last fiddled with by 3.14159 on 2010-09-09 at 16:25

2010-09-09, 16:30	#455
3.14159 May 2010 Prime hunting commission. 2⁴×3×5×7 Posts	Actually, it is 3 * 2^5082306 + 1. (1529928 digits) Never specified anything against 321 primes. The largest 2 * 3ⁿ +1 prime is where n = 1175232, apparently, at about 560729 digits. Last fiddled with by 3.14159 on 2010-09-09 at 16:35

2010-09-09, 16:42	#456
CRGreathouse Aug 2006 3×1,993 Posts	Good call. My search was for Code: ^[23]^%[23]^%+1$ but I should have also checked Code: ^[23][23]^%+1$ and Code: ^[23]^%+1$ (not that more Fermat primes are likely...) Of course technically the % is over-broad, but [[:digit:]]{1,} is so verbose...! Last fiddled with by CRGreathouse on 2010-09-09 at 16:45

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Prime posting thread, part 2. (With a catch.)	3.14159	Miscellaneous Math	55	2010-11-19 23:55
Tiny range request .... 555.1M	petrw1	LMH > 100M	1	2010-07-13 15:35
Other primes thread	nuggetprime	No Prime Left Behind	32	2009-10-21 21:48
Error: tiny factoring failed	10metreh	Msieve	26	2009-03-08 23:28
Tiny error on nfsnet pages.	antiroach	NFSNET Discussion	1	2003-07-08 00:27

2010-09-10, 00:10	#461
kar_bon Mar 2006 Germany 2²×727 Posts	... and 10^999-6251 is the next smaller prime with 999 digits.