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:

55296208019546011056775958772365230026104973043748330012177931228756932285450242993933010742357143453482970919578601914425364238598294955738400021039217889466218429690776919792392100099253267857786026933823079081872664185916619928957134040681854089954501328861790047337839438968641009129328048212598613711518471458100654402707671206595091614716963872127792522912877973266672906659893955040225811971790734939791477041412519720089212143550977004410489480134640312059102009267824231125401189679223454319120430478532646641861679796970158601277674150305960796516408256001389952197460071104523287187057939857578894288385161206913777361215347305711699166626856247007575807475549532084099872581200028647350841877515058697373416079953566825794626286250291911581453465199943164456865786532143392254253890977853635461707468273799851855677302030287198063631196021856754735139760262207607740738635184398089845320017721206060926880502934361830466788983738238437319687690998175899430298290976037491899545962048709816541173489830780412288240036226559391156983933161699233872164300764124657511298452125102713252631493091849422290342944145569963529933649502302112325743422012828662298587555600835989389269791045707097701867138242034880801778477304166493923065738098403485321051351703784971266054613937138301255267925587958496881194968278414571794269583465075924266371568038878481174849786884077124308593263537530889040182914424436260972523244792018184946966332476272653209351545273637152458935129228010906419629303323329188681239923527519419346667461002682797817623725255242063815454313565465221302542096243754253539293306843387501412792697781382910900898092009990848422411568415365937731667891740606993384697944206642302560846209081425651212929325549402744733580715783893458935758174082391024205035146401685507403570454133675330769645693435886718138166385500326088192273941886949321232529271266515584489070042414443108600836017144503089358180955384423010410758362466553109259612303358612607340084478716323146669011667848301518737695155695068035733537640877704654471853741947366673625097639146219077618180214314022085635619635759311455698165518164425088302248800665249155797051129521992215357937443136892147806807361777889507320811777237656448865422301923194309848039409072014476444306986449655117895071668056542049184200561391634419560666320666624975150313099327282561830475220336047127539838249543895284125317429185448677741163671818546547999344911467479081576194801560022982483401162600844402129882461290016286086147573426137374594855642444631188837852047949678355730027225602815408205243641185507004290671109233770347567908298916007624772456430908447533819052834460270186700643250311906533830048504110826639044886711622852202477905057125214208836607796394966943502875389359204425627407590155386948047101033899449710609513030320630594126015078003503032621646799238956646089699783779248587146562480910117140675854436237131466365834539957961193344839935864641616192029990914624531836587087442358413042418830544136779552215435220537585853798020172635701556636879569980970657652334508018404247167441561989354968717252502259497245452701016678125415533660816994318112155880145928059484101565848503039489171307299687419735973800712064914745796377044600050871000728971367893752758814311181052681841868478290290493341439146213535353534974411513555553667105438299862520854335883076221088390896147765959317546855755645731112916110852249266961900110419733040680937635333126762436610172752893142117595456872950988404344318281288779760311130030263642365211149492255296548795193491017664061171050552355193542783994674067989274258615978269876016401221686578931808240688217137622614400561781115113964489299565040763836605282276416799263673315165481138595568718189547088851512974647092195738066490686576038348974556454903016090678053328318861570202505768117513695086217686608309052975949633767321320619513872908179572572932702412163854133429123516682754693554023228141329633542202879887778213879645535308453416970013859017542079230477234826112177889759024995092082283435966879498086128494144239712003861528973252021124510859723803924121155457156291521235513551892582905510497239388296627720417532312437846572250413611094845810063336747739801030080710566130762782124878608067231304712477235423773985994233015987194552829205073410633776242062417213109569563719885555974742701909474889617598378612649829645843308049466953471661523775396099039294611868694236422973186652845198492114219991932794809808881234029589499866590515853023525653061832261427496428732603057419490394252622466343902056345132082627937868596717147880585324510093478061772987256374539995241255939419301237049258892751149725408376877270817057001056820747903788067514917750408300696934324078490667046169168882217810171710169596373516726484658891338274712828588856042452155268041433774856018995706514300093692912300779969016766555666931569077291462502268798879420518802489736998153294630762419042707721216522367565622272724845802044063419243210961411807865703218245206247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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. 2⁴·3·5·7 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. 690₁₆ 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 1011101011011₂ 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

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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.