Prime-generating polynomials - Page 2

HiddenWarrior · 2003-10-21, 13:53

1..27: Y=[935722367*X^26+-342980172535*X^25+59705244800500*X^24+-6567330851190850*X^23+512324805492067165*X^22+-30159045300454180825*X^21+1391689317707546464750*X^20+-51623777158001343895600*X^19+1566255559997226681842545*X^18+-39340574890944019531921225*X^17+824951972735456935907785000*X^16+-14522488302656507710385841850*X^15+215325390496915856059658076955*X^14+-2692453605856953638557928754775*X^13+28376910178755788686487904397750*X^12+-251497329469943024139664495746100*X^11+1866492021756803627460959861707960*X^10+-11526806034118926830897311127090800*X^9+58718517650596092518514090107588000*X^8+-243824340833695667450656545039345600*X^7+812262111914934010939105118794843008*X^6+-2124380386387287271385586557767779840*X^5+4232225658073134471239727803614464000*X^4+-6145242694018853176115544652692480000*X^3+6066229867902412851711630741258240000*X^2+-3591286494253211632003175315865600000*X^1+945183686053782824268934938624000000]/10888869450418352160768000000
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HiddenWarrior · 2003-10-21, 13:54

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HiddenWarrior · 2003-10-21, 13:55

1..38: Y=[-22187604356629*X^37+16047306933609124*X^36+-5595137388109054599*X^35+1252649728181807248020*X^34+-202367772502890768307632*X^33+25134932503469122460897232*X^32+-2497131579499019997559664012*X^31+203849142504685478714978453840*X^30+-13939332169795198604832385354374*X^29+809932290105475832131652644886904*X^28+-40425043151417600893319046906947874*X^27+1747799786294861747606138920823697240*X^26+-65887661788624003805661684704302105260*X^25+2176627318557032603578295387441531510160*X^24+-63257826743415569093836608519052850968860*X^23+1621991887414091152286117057637611018778000*X^22+-36768191522602808513859274731311050709387565*X^21+737805828944311902379848808744333515464366340*X^20+-13113368568309162414849391479014872327733729615*X^19+206423786863823646108982496218299388911137458100*X^18+-2875874835925288304411210357195182040781752756316*X^17+35412258510862669189124166004512584172891979840896*X^16+-384612791852695825527328142856308897164997650022496*X^15+3674300803833122112160528301900949657105068650769280*X^14+-30764313814138937229170169100530655839054138063741568*X^13+224738661549667564413580927324059384959103353778976768*X^12+-1424387624452577925925735492572433222560346739197625088*X^11+7778346188447952912419017580010371662785602827565429760*X^10+-36285455377997366382053736524998302298149154670516038656*X^9+143064590397962434964886921812209625319690243824299032576*X^8+-470383700688832433611394424628521131121802432514304147456*X^7+1267681154771522082304257602070379855567174603926251765760*X^6+-2737497995390424705552683004493464183418055167673597952000*X^5+4592260663895229023720896832393587922670025156138106880000*X^4+-5723214736791975164534128643723346322514145028173987840000*X^3+4941880772850779568876477643461949500717492429284966400000*X^2+-2608823088418553245182828917713172215938322268160000000000*X^1+624766539422177757145962278107222512957032654438400000000]/523022617466601111760007224100074291200000000
1..39: Y=[35469146336355*X^38+-27125766400790957*X^37+10010220986147648717*X^36+-2374388432433991587567*X^35+406830411438609446733900*X^34+-53652156199833215114608056*X^33+5666369427422718216302803956*X^32+-492350984217010392809580757996*X^31+35883412014210842494056304733050*X^30+-2225400015277235686211436721845942*X^29+118735768641902697182541976071913782*X^28+-5496742429723315142952044594349095442*X^27+222261048352299393452909478478815601020*X^26+-7890595697364956494573641846438163859580*X^25+246940439012656170871841326801506899949780*X^24+-6833411838723001553264714913637454766832380*X^23+167576581851550575495922340588508592346559675*X^22+-3647288260903478028312001547641454440449564645*X^21+70513295096318178493855908989470360443336823845*X^20+-1211183652340733881884517642676735135425209066295*X^19+18476552282351056856396162869283883588644851042720*X^18+-250079778999200718297301858697685087751549980756228*X^17+2998412891097690541299317960013376443781315392119568*X^16+-31775041743201212389605231198106308915786385337273568*X^15+296744097293038973088919758166045335258229626911328000*X^14+-2433070069491807328094004471772947486463201019552092544*X^13+17433439470206514191151914809817016472631557135221196544*X^12+-108537363535953180183350212929659218149716856479421506304*X^11+583026393804568154076485483306447322084547252074177945600*X^10+-2678855017388399654355572348778690259275862383527571330048*X^9+10415905534057594673980666386482065751032528256771174903808*X^8+-33811883718372990814251272839334281228876488337090765160448*X^7+90065666928766371820004801990514700577072175555792904519680*X^6+-192440095758955612630650280549849112008211385113696305152000*X^5+319747062067704265632507800804089529937037378837085552640000*X^4+-395089206493001205426490284814180250672343180527240478720000*X^3+338577143477903297645645316791730609983063561580262195200000*X^2+-177567788409291714663748629092011115517101121302691840000000*X^1+42292294254189052226303196706771938999967044285235200000000]/20397882081197443358640281739902897356800000000

All proves can be found on my new book that will soon be available "Mersenne's Building-Equations and Koefficients" at my site http://www.yxine.km.ru/mpf

ixfd64 · 2003-10-21, 18:05

Is there a program that finds those polynomials? If there is one, I'd sure like to know where to download it.

HiddenWarrior · 2003-10-22, 09:55

This must work:

http://www.yxine.km.ru/zip/mbe.zip

Still many errors!

koal · 2003-10-22, 11:16

Seems so! Looking at the first few of your polynomials i get some strange results:

===> x=1;Y=(-1*X**3+9*X**2+-14*X+18)/24;say y
===> 0.5

===> x=2;Y=(-1*X**3+9*X**2+-14*X+18)/24;say y
===> 0.75

===> x=3;Y=(-1*X**3+9*X**2+-14*X+18)/24;say y
===> 1.25

===> x=4;Y=(-1*X**3+9*X**2+-14*X+18)/24;say y
===> 1.75

What did you mean with 1..4 ?

HiddenWarrior · 2003-10-22, 11:38

Hell yeah! There was a mistake in my records. As you can check the divider is wrong!!! The program gives right results. So use it to build right polynominals.

koal · 2003-10-22, 13:29

But with the correct divisor of "6" you get the first four regular primes 2, 3, 5, 7

Where are the mersennes? Did you read my thread at

http://www.mersenneforum.org/showthr...=&threadid=925 ?

Aren't your polynomials a bit like that?

HiddenWarrior · 2003-10-23, 06:29

Quote:

Originally posted by koal
But with the correct divisor of "6" you get the first four regular primes 2, 3, 5, 7

Where are the mersennes? Did you read my thread at

Ok, You have to take polynominal 1..39:

Y=[35469146336355*X^38-27125766400790957*X^37+10010220986147648717*X^36-2374388432433991587567*X^35+406830411438609446733900*X^34-53652156199833215114608056*X^33+5666369427422718216302803956*X^32-492350984217010392809580757996*X^31+35883412014210842494056304733050*X^30-2225400015277235686211436721845942*X^29+118735768641902697182541976071913782*X^28-5496742429723315142952044594349095442*X^27+222261048352299393452909478478815601020*X^26-7890595697364956494573641846438163859580*X^25+246940439012656170871841326801506899949780*X^24-6833411838723001553264714913637454766832380*X^23+167576581851550575495922340588508592346559675*X^22-3647288260903478028312001547641454440449564645*X^21+70513295096318178493855908989470360443336823845*X^20-1211183652340733881884517642676735135425209066295*X^19+18476552282351056856396162869283883588644851042720*X^18-250079778999200718297301858697685087751549980756228*X^17+2998412891097690541299317960013376443781315392119568*X^16-31775041743201212389605231198106308915786385337273568*X^15+296744097293038973088919758166045335258229626911328000*X^14-2433070069491807328094004471772947486463201019552092544*X^13+17433439470206514191151914809817016472631557135221196544*X^12-108537363535953180183350212929659218149716856479421506304*X^11+583026393804568154076485483306447322084547252074177945600*X^10-2678855017388399654355572348778690259275862383527571330048*X^9+10415905534057594673980666386482065751032528256771174903808*X^8-33811883718372990814251272839334281228876488337090765160448*X^7+90065666928766371820004801990514700577072175555792904519680*X^6-192440095758955612630650280549849112008211385113696305152000*X^5+319747062067704265632507800804089529937037378837085552640000*X^4-395089206493001205426490284814180250672343180527240478720000*X^3+338577143477903297645645316791730609983063561580262195200000*X^2-177567788409291714663748629092011115517101121302691840000000*X+42292294254189052226303196706771938999967044285235200000000]/523022617466601111760007224100074291200000000

When you put natural numbers 1..39 instead of x, you'll get first 39 Mersenne primes. It is the main idea of this polynominal. Further more, as we use LaGrange interpolation to build them, we can replace x with 40 and we'll get "something like 40th Mersenne prime". Ofcource this value will be with a great error-rate, coz Mersenne primes are strongly intropiated (and LaGrange interpolation works great only with linear values). So, the only (?) way to us to predict 40th MP is to find right koefficients for polynominal 1..40. To build it we must find 40 koefficients. We know for sure all previous 39 polynominals, divider and signs of koefficients. To find koefficients we must find some connections between them. So it is another idea of this stuff!

mfgoode · 2004-03-01, 18:30

Quote:

Originally Posted by ixfd64

I've read in a book that f(x) = x² + x + 41 is a very good prime-generating polynomial.

From Memory I give you a better one- n^2-79n+1601. Please check it out
Mally

wpolly · 2004-03-02, 12:57

Quote:

Originally Posted by TravisT

Could someone give me the details of the proof that there is no polynomial with integer coefficients that generates only primes? I think it was Goldbach who proved it, but I can't find any details of the proof online.

I asked someone about it and he said that you use the fact that f(f(0)) is divisible by f(0), but what if f(f(0)) = f(0)?

let f(0)=p be a prime.
Obviously, f(kp)=f(0)=0(mod p), so p|f(kp).
Since f is a Prime-Generating Polynomial, f(kp) can only be p, 0, or -p.
But there's at most 3n values of k which f(kp)=0,p,-p(n is the degree of f)
Contradiction.

Q. E. D.

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2003-10-21, 13:53	#12
HiddenWarrior Jun 2003 Russia, Novosibirsk 2·107 Posts	1..27: Y=[935722367X^26+-342980172535X^25+59705244800500X^24+-6567330851190850X^23+512324805492067165X^22+-30159045300454180825X^21+1391689317707546464750X^20+-51623777158001343895600X^19+1566255559997226681842545X^18+-39340574890944019531921225X^17+824951972735456935907785000X^16+-14522488302656507710385841850X^15+215325390496915856059658076955X^14+-2692453605856953638557928754775X^13+28376910178755788686487904397750X^12+-251497329469943024139664495746100X^11+1866492021756803627460959861707960X^10+-11526806034118926830897311127090800X^9+58718517650596092518514090107588000X^8+-243824340833695667450656545039345600X^7+812262111914934010939105118794843008X^6+-2124380386387287271385586557767779840X^5+4232225658073134471239727803614464000X^4+-6145242694018853176115544652692480000X^3+6066229867902412851711630741258240000X^2+-3591286494253211632003175315865600000X^1+945183686053782824268934938624000000]/10888869450418352160768000000 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1..29: 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1..30: 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2003-10-21, 13:54	#13
HiddenWarrior Jun 2003 Russia, Novosibirsk 2×107 Posts	1..34: 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2003-10-21, 13:55	#14
HiddenWarrior Jun 2003 Russia, Novosibirsk 2·107 Posts	1..38: Y=[-22187604356629X^37+16047306933609124X^36+-5595137388109054599X^35+1252649728181807248020X^34+-202367772502890768307632X^33+25134932503469122460897232X^32+-2497131579499019997559664012X^31+203849142504685478714978453840X^30+-13939332169795198604832385354374X^29+809932290105475832131652644886904X^28+-40425043151417600893319046906947874X^27+1747799786294861747606138920823697240X^26+-65887661788624003805661684704302105260X^25+2176627318557032603578295387441531510160X^24+-63257826743415569093836608519052850968860X^23+1621991887414091152286117057637611018778000X^22+-36768191522602808513859274731311050709387565X^21+737805828944311902379848808744333515464366340X^20+-13113368568309162414849391479014872327733729615X^19+206423786863823646108982496218299388911137458100X^18+-2875874835925288304411210357195182040781752756316X^17+35412258510862669189124166004512584172891979840896X^16+-384612791852695825527328142856308897164997650022496X^15+3674300803833122112160528301900949657105068650769280X^14+-30764313814138937229170169100530655839054138063741568X^13+224738661549667564413580927324059384959103353778976768X^12+-1424387624452577925925735492572433222560346739197625088X^11+7778346188447952912419017580010371662785602827565429760X^10+-36285455377997366382053736524998302298149154670516038656X^9+143064590397962434964886921812209625319690243824299032576X^8+-470383700688832433611394424628521131121802432514304147456X^7+1267681154771522082304257602070379855567174603926251765760X^6+-2737497995390424705552683004493464183418055167673597952000X^5+4592260663895229023720896832393587922670025156138106880000X^4+-5723214736791975164534128643723346322514145028173987840000X^3+4941880772850779568876477643461949500717492429284966400000X^2+-2608823088418553245182828917713172215938322268160000000000X^1+624766539422177757145962278107222512957032654438400000000]/523022617466601111760007224100074291200000000 1..39: Y=[35469146336355X^38+-27125766400790957X^37+10010220986147648717X^36+-2374388432433991587567X^35+406830411438609446733900X^34+-53652156199833215114608056X^33+5666369427422718216302803956X^32+-492350984217010392809580757996X^31+35883412014210842494056304733050X^30+-2225400015277235686211436721845942X^29+118735768641902697182541976071913782X^28+-5496742429723315142952044594349095442X^27+222261048352299393452909478478815601020X^26+-7890595697364956494573641846438163859580X^25+246940439012656170871841326801506899949780X^24+-6833411838723001553264714913637454766832380X^23+167576581851550575495922340588508592346559675X^22+-3647288260903478028312001547641454440449564645X^21+70513295096318178493855908989470360443336823845X^20+-1211183652340733881884517642676735135425209066295X^19+18476552282351056856396162869283883588644851042720X^18+-250079778999200718297301858697685087751549980756228X^17+2998412891097690541299317960013376443781315392119568X^16+-31775041743201212389605231198106308915786385337273568X^15+296744097293038973088919758166045335258229626911328000X^14+-2433070069491807328094004471772947486463201019552092544X^13+17433439470206514191151914809817016472631557135221196544X^12+-108537363535953180183350212929659218149716856479421506304X^11+583026393804568154076485483306447322084547252074177945600X^10+-2678855017388399654355572348778690259275862383527571330048X^9+10415905534057594673980666386482065751032528256771174903808X^8+-33811883718372990814251272839334281228876488337090765160448X^7+90065666928766371820004801990514700577072175555792904519680X^6+-192440095758955612630650280549849112008211385113696305152000X^5+319747062067704265632507800804089529937037378837085552640000X^4+-395089206493001205426490284814180250672343180527240478720000X^3+338577143477903297645645316791730609983063561580262195200000X^2+-177567788409291714663748629092011115517101121302691840000000*X^1+42292294254189052226303196706771938999967044285235200000000]/20397882081197443358640281739902897356800000000 All proves can be found on my new book that will soon be available "Mersenne's Building-Equations and Koefficients" at my site http://www.yxine.km.ru/mpf

2003-10-21, 18:05	#15
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 2³·13·23 Posts	Is there a program that finds those polynomials? If there is one, I'd sure like to know where to download it.

2003-10-22, 09:55	#16
HiddenWarrior Jun 2003 Russia, Novosibirsk D6₁₆ Posts	This must work: http://www.yxine.km.ru/zip/mbe.zip Still many errors!

2003-10-22, 11:16	#17
koal Nov 2002 Vienna, Austria 41 Posts	Seems so! Looking at the first few of your polynomials i get some strange results: ===> x=1;Y=(-1X3+9X*2+-14X+18)/24;say y ===> 0.5 ===> x=2;Y=(-1X3+9X*2+-14X+18)/24;say y ===> 0.75 ===> x=3;Y=(-1X3+9X*2+-14X+18)/24;say y ===> 1.25 ===> x=4;Y=(-1X3+9X*2+-14X+18)/24;say y ===> 1.75 What did you mean with 1..4 ?

2003-10-22, 11:38	#18
HiddenWarrior Jun 2003 Russia, Novosibirsk 2·107 Posts	Hell yeah! There was a mistake in my records. As you can check the divider is wrong!!! The program gives right results. So use it to build right polynominals.

2003-10-22, 13:29	#19
koal Nov 2002 Vienna, Austria 51₈ Posts	But with the correct divisor of "6" you get the first four regular primes 2, 3, 5, 7 Where are the mersennes? Did you read my thread at http://www.mersenneforum.org/showthr...=&threadid=925 ? Aren't your polynomials a bit like that?