Unders\303\270gende aktivitet om primtal.Af Petur Birgir PetersenDefinition:Et primtal er et naturligt tal st\303\270rre end 1, som kun 1 og tallet selv g\303\245r op i.Eksempel 1: Tallet 1 ikke et primtal fordi det ikke er st\303\270rre end 1.Tallet 2 er et primtal, fordi 2 er et naturligt tal forskelligt fra 1, som kun 1 og 2 g\303\245r op i.Tallet 3 er et primtal, fordi 3 er et naturligt tal forskelligt fra 1, som kun 1 og 3 g\303\245r op i.Tallet 4 er ikke et primtal, fordi 1 og 2 og 4 g\303\245r op i 4.Erathostenes' si:Erathosthenes' si er en antik gr\303\246sk metode til at lave lister over primtal op til en vis gr\303\246nse.Man begynder med at skrive de naturlige tal st\303\270rre end 1 og op til et tal n p\303\245 en liste, markerer det mindste tal 2 og sletter alle de tal p\303\245 listen, som 2 g\303\245r op i. I de tal, der nu er tilbage, markeres det mindste tal 3,og alle de tal p\303\245 listen, som 3 g\303\245r op i, slettes . I de tal, som derefter er tilbage, markeres det mindste tal 5,og alle de tal p\303\245 listen, som 5 g\303\245r op i, slettes .S\303\245dan forts\303\246ttes til der kun er primtal tilbage p\303\245 listen.Bem\303\246rk, at ovenst\303\245ende algoritme kan stoppes, n\303\245r man markerer et primtal p, som er st\303\270rre end eller lig med kvadratroden af n .Forklar dette! Opgave 1:Brug Erathostenes' si til at slette de tal i f\303\270lgende tabel, der ikke er primtal:
Primtal i Maple:N\303\245r man skal i gang med at arbejde med primtal i Maple, er det lettest, hvis man f\303\270rst aktiverer den indbyggede pakke NumberTheory (husk at Maple skelner mellem sm\303\245 og store bogstaver):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRLU51bWJlclRoZW9yeUYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==Egentlig beh\303\270ves man kun at aktivere pakken, hvis man har t\303\246nkt sig at bruge en af kommandoerne Divisors, NumberOfPrimefactors, PrimeCounting eller PrimeFactors.De andre kommandoer vi bruger erindbyggede, og virker selv om vi ikke bruger pakken NumberTheory.Hvis man ikke er interesseret i at se de mulige kommandoer fra pakken, afsluttes hentningen af pakken med et semikolon:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRLU51bWJlclRoZW9yeUYnRi9GMi9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI6RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZUFor at lave en liste over alle primtal, som er mindre eller lig med et naturligt tal n , kan man g\303\270re s\303\245ledes:Find f\303\270rst ud af, hvor mange primtal, det drejer sig om. Dette g\303\270res ved hj\303\246lp af kommandoen PrimeCounting.Brug derefter de indbyggede kommandoer seq og ithprime til at lave listen. Kommandoen seq laver en sekvens, og ithprime giver primtal nummer i.Man kan f\303\245 en vejledning til de enkelte kommandoer ved i matematikmode at skrive ? foran kommandoen f.eks.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEkc2VxRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdGNEYvI stedet for at l\303\246se hele hj\303\246lpedokumentet, er det en god ide at begynde med at se p\303\245 eksemplerne nederst i dokumentet.Eksempel 2:Man kan lave en liste over alle primtal, som er mindre eller lig med 100 s\303\245ledes:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEuUHJpbWVDb3VudGluZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEkMTAwRicvRjNRJ25vcm1hbEYnRj4tSSNtb0dGJDYtUSI7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==Output viser, at der er 25 primtal mellem 1 og 100.Dem kan man s\303\245 finde et efter et med kommandoen ithprime: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEpaXRocHJpbWVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiMtSSNtbkdGJDYlUSIzRidGMi9GNlEnbm9ybWFsRidGMkZBLUkjbW9HRiQ2LlEiO0YnRjJGQS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==Men dette g\303\245r jo meget langsomt, s\303\245 det er her at kommandoen seq kommer os til hj\303\246lp. Eftersom vi ved, at vi skal finde primtal nummer 1 til og med 25, skriver vi:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkc2VxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiotRiw2JVEpaXRocHJpbWVGJ0YvRjItRjY2JC1GIzYjLUYsNiVRImlGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRkQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGQS1GRzYtUSI9RidGREZKL0ZORkxGT0ZRRlNGVUZXL0ZaUSwwLjI3Nzc3NzhlbUYnL0ZnbkZeby1JI21uR0YkNiRRIjFGJ0ZELUZHNi1RIy4uRidGREZKRlxvRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GZW4tRmFvNiRRIzI1RidGREZELUZHNi1RIjtGJ0ZERkpGTUZPRlFGU0ZVRldGWUZfbw==Alt dette kan komprimeres til den ene kommandolinje: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Opgave 2:Brug Maple til at lave lister over:a) Alle primtal, som er mindre eller lig med 1000.b) Alle primtal, som er mindre eller lig med 10000.c) Alle primtal, som er mindre eller lig med 100000.d) Alle primtal, som er mindre eller lig med 1000000.Den sidste af disse lister blev s\303\245 stor, at den var helt uoverskuelig, og s\303\245 skulle man ogs\303\245 vente i ret lang tid, f\303\270r Maple blev f\303\246rdig med beregningerne!Husk altid at gemme dit arbejde inden du s\303\246tter store udregninger i gang. Det er muligt at afbryde en igangv\303\246rende udregning ved at venstreklikke p\303\245 knappen i menulinjen indeholdende et udr\303\245bstegn inde i en ottekant. Hvis udregningen ikke stopper indenfor en rimelig tidshorisont, kan det blive n\303\270dvendigt at lukke dokumentet, og \303\245bne det igen.Hvis man f.eks. kun er interesseret i primtallene i et givent interval f.eks. primtallene mellem 1000000 og 1001000, kan man skrive s\303\245ledes:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEkc2VxRicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYuLUYsNiZRKWl0aHByaW1lRidGL0YyRjUtRjk2JS1GIzYjLUYsNiZRImlGJ0YvRjJGNUYyL0Y2USdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0YyRkcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGTy8lKnN5bW1ldHJpY0dGTy8lKGxhcmdlb3BHRk8vJS5tb3ZhYmxlbGltaXRzR0ZPLyUnYWNjZW50R0ZPLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGRC1GSjYuUSI9RidGMkZHRk0vRlFGT0ZSRlRGVkZYRlovRmduUSwwLjI3Nzc3NzhlbUYnL0ZqbkZhby1GLDYmUS5QcmltZUNvdW50aW5nRidGL0YyRjUtRjk2JS1GIzYjLUkjbW5HRiQ2JVEoMTAwMDAwMEYnRjJGR0YyRkctRko2LlEiK0YnRjJGR0ZNRl9vRlJGVEZWRlhGWi9GZ25RLDAuMjIyMjIyMmVtRicvRmpuRmJwLUZbcDYlUSIxRidGMkZHLUZKNi5RIy4uRidGMkZHRk1GX29GUkZURlZGWEZaRmFwL0ZqbkZobkZjby1GOTYlLUYjNiMtRltwNiVRKDEwMDAxMDBGJ0YyRkdGMkZHRjJGRy1GSjYuUSI7RidGMkZHRk1GUEZSRlRGVkZYRlpGZm5GYm8=Som man kan se, er der f\303\246rre primtal mellem 1 og 100, end der er mellem 1000000 og 1000100. Dette kan ses, ved at t\303\246lle tallene i listen, men det er hurtigere og lettere at skrive:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEuUHJpbWVDb3VudGluZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEkMTAwRicvRjNRJ25vcm1hbEYnRj4tSSNtb0dGJDYtUSI7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==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Opgave 3:a) Find antallet af primtal, som er mindre eller lig med 1000.b) Find antallet af primtal, som er mindre eller lig med 10000.c) Find antallet af primtal, som er mindre eller lig med 100000.d) Find antallet af primtal, som er mindre eller lig med 1000000.e) Find antallet af primtal, som er mindre eller lig med 10000000.f) Find antallet af primtal, som er mindre eller lig med 100000000.Nu begyndte udregningen at g\303\245 langsomt her til sidst.Man kan n\303\270jes med at betragte intervaller svarende til f.eks. 1000: Opgave 4:a) Find antallet af primtal mellem 10000 og 11000.b) Find antallet af primtal mellem 100000 og 101000.c) Find antallet af primtal mellem 1000000 og 1001000.d) Find antallet af primtal mellem 10000000 og 10001000.e) Find antallet af primtal mellem 100000000 og 100001000.Hvis man fors\303\270ger at forts\303\246tte med endnu st\303\270rre tal begynder det at g\303\245 lidt langsomt med Maples udregning.Forklaringen er brugen af kommandoen PrimeCounting.Det g\303\245r hurtigere, hvis man n\303\270jes med at unders\303\270ge tallene i et givent interval med en sammensat kommando: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 ord kan denne kommando beskrives s\303\245ledes:antallet af elementer udvalgt blandt primtallene i intervallet [1000000000; 1000001000].Vi indf\303\270rer nu funktionen \317\200, der t\303\246ller antallet af primtal mindre eller lig med et naturligt tal n , s\303\245ledes:\317\200(n) = antallet af primtal mindre end eller lig med n.Opgave 5:Brug resultaterne fra opgave 3 til at finde \317\200(1000), \317\200(10000), \317\200(100000), \317\200(1000000), \317\200(10000000) og \317\200(100000000). Chebyshevs s\303\246tning, der blev bevist i midten af det 19. \303\245rhundrede, siger f\303\270lgende om t\303\246llefunktionen \317\200:For store naturlige tal n er 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For eksempel har vi for n =10000 at 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og 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Eftersom 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s\303\245 er tallet n =10000 \303\245benbart for lille til at s\303\246tningen g\303\246lder for dette tal.Opgave 6:Unders\303\270g om s\303\246tningen g\303\246lder for:a) n =100000b) n =1000000c) n =10000000d) n =100000000 Opgave 7:Find andelen af primtal i procent af a) De naturlige tal mindre end 100000b) De naturlige tal mindre end 1000000c) De naturlige tal mindre end 10000000d) De naturlige tal mindre end 10000000e) G\303\270r ud fra Chebyshevs s\303\246tning rede for, at andelen af primtal i procent af naturlige tal mindre end n vil v\303\246re aftagende n\303\245r n vokser.Primtalstvillinger er to primtal st\303\270rre end to, hvis differens er to.For eksempel er 17 og 19 primtalstvillinger.Selv om den gennemsnitlige afstand mellem primtallene i f\303\270lge Chebyshevs s\303\246tning vokser, formoder man at der findes uendeligt mange primtalstvillinger.Dette er ikke en s\303\246tning, men en formodning, fordi ingen har form\303\245et at bevise (eller modbevise) dette.I den n\303\246ste opgave skal vi lede efter primtalstvillinger. Men f\303\270rst et eksempel:Eksempel 3:Lad os se p\303\245 100 primtal ad gangen f.eks. fra nummer 10001 til og med 10100: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Man kan faktisk finde primtalstvillinger inde i listen, men man skal kigge godt efter for at opdage dem. Pr\303\270v finde nogle af disse primtalstvillinger.Hvis vi kun er interesseret i at unders\303\270ge om der findes primtalstvillinger i intervallet, er det mere overskueligt at udregne forskellen mellem to naboprimtal: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Ja, der er en del totaller og dermed ogs\303\245 primtalstvillinger p\303\245 listen. Hvis vi vil t\303\246lle antallet af primtalstvillinger p\303\245 listen, er det mere overskueligt at sortere denne f\303\270rst: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Som man kan se var der hele 8 primtalstvillinger mellem primtal med nummer 10000 til og med 10100.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=opgave 8:Find antallet af primtalstvillinger mellem primtal med nummer:a) 1 til og med 100b) 1001 til og med 1100c) 100001 til og med 100100d) 1000001 til og med 1000100e) 10000001 til og med 10000100f) 20000001 til og med 20000100Den franske matematiker Fermat havde en formodning om at tal, der udregnes efter formlen: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alle er primtal.Lad os udregne de f\303\270rste 8 af disse s\303\245kaldte Fermat-tal: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For at unders\303\270ge om disse Fermat-tal er primtal, kan man i Maple bruge kommandoen isprime f.eks.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 vi fik at vide var at det er sandt at 17 er et primtal, men at det ikke er sandt at 18 er et primtal.For at finde ud af hvilke naturlige tal, der s\303\245 g\303\245r op i 18 kan man f.eks. bruge kommandoen Divisors s\303\245ledes:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEpRGl2aXNvcnNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1JI21uR0YkNiRRIzE4RicvRjNRJ25vcm1hbEYnRj4tSSNtb0dGJDYtUSI7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==For at faktorisere tallet i dets primfaktorer kan man bruge kommandoen ifactor, hvor bogstavet i st\303\245r for integer:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoaWZhY3RvckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEjMThGJy9GM1Enbm9ybWFsRidGPi1JI21vR0YkNi1RIjtGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnDette skal l\303\246ses s\303\245ledes at 18 kan skrives som LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGL0YyRkovJStleGVjdXRhYmxlR0Y4Ri8=. Opgave 9:Unders\303\270g om alle de f\303\270rste 8 Fermat-tal er primtal.Hvis et (eller flere) af disse tal ikke er et primtal, s\303\245 a) Find ud af hvilke naturlige tal, der g\303\245r op i de p\303\245g\303\246ldende Fermat-tal b) Faktoriser tallet i dets primfaktorer. Som man hurtigt opdager, s\303\245 bliver Maples udregninger langsomme, n\303\245r de primtal,som vi regner p\303\245 bliver for store.Opgave 10: Forklar hvorfor det med brug af Erathostenes si vil tage l\303\246ngere tid at unders\303\270ge om et tal er et primtal, jo st\303\270rre tallet er. I antikken udarbejdede Euklid f\303\270lgende bevis for at der eksisterer uendelig mange primtal.I beviset bruger Euklid Aritmetikkens Fundamentals\303\246tning, som siger, at ethvert tal p\303\245 en entydig m\303\245de (bortset fra r\303\246kkef\303\270lgen) kan faktoriseres i primfaktorer.Skitse af Euklids bevis for at der eksisterer uendelig mange primtal:Lad 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 v\303\246re primtal. S\303\245 g\303\245r ingen af disse primtal op i tallet
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdWJHRiQ2JS1GLDYlUSJwRidGL0YyLUYjNictSSNtbkdGJDYkUSIxRidGOUYvLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGMUYyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORmFvLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGYG9GYm8tRlA2JUZSLUYjNiktRlg2JVEiMkYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y5Ri9GZW4vJStleGVjdXRhYmxlR0Y9RmhuL0ZecFElVGV4dEYnRjJGam4tRjY2LkZfb0ZdcEY5RjtGPkZARkJGREZGRkhGYG9GYm8tRjY2LkZlb0ZdcEY5RjtGPkZARkJGREZGRkhGYG9GYm8tRjY2LlEjLi5GJ0ZdcEY5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRidGYm9GaHAtRjY2LlEiLkYnRl1wRjlGO0Y+RkBGQkZERkZGSEZgb0Zib0ZmcEZkcEZmcC1GUDYlLUYsNiZGVEYvRl1wRjItRiM2KS1GLDYmUSJqRidGL0ZdcEYyRi9GZW5GYHBGaG5GYnBGMkZqbkZgcEY5 +1
(idet dets rest ved division med ethvert af primtallene 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 er 1). Primfaktorerne i n, som jo findes if\303\270lge Aritmetikkens Fundamentals\303\246tning, kan alts\303\245
ikke findes blandt tallene 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 , s\303\245 de er alts\303\245 andre (og dermed nye) primtal.
Heraf f\303\270lger umiddelbart, at der m\303\245 v\303\246re uendeligt mange primtal.Eksempel 4:Euklids bevis for at der eksisterer uendelig mange primtal, kan bruges til at fremstille en liste af primtal:Antag f.eks. at vi indtil videre kun kender primtallet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGNEY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Y+= 2. Det vil sige at primtalslisten til at begynde med kun best\303\245r af tallet 2.Nu beregnes tallet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGNEY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+ +1 = 2+1 = 3, som er et nyt primtal 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.Derefter beregnes tallet 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 +1 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGLy1GMzYtUSIrRidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSwwLjIyMjIyMjJlbUYnL0ZJRlEtRiw2JFEiMUYnRi8vJStleGVjdXRhYmxlR0Y4Ri8= = 7, som er et nyt primtal 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.Derefter beregnes tallet 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 +1 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGL0YyLUYsNiRRIjdGJ0YvLUYzNi1RIitGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGVC1GLDYkUSIxRidGLy8lK2V4ZWN1dGFibGVHRjhGLw== = 43, som er et nyt primtal 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. Derefter beregnes tallet 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 +1 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGL0YyLUYsNiRRIjdGJ0YvRjItRiw2JFEjNDNGJ0YvLUYzNi1RIitGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGVy1GLDYkUSIxRidGLy8lK2V4ZWN1dGFibGVHRjhGLw== = 1807LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=. Allerede nu kan man ikke umiddelbart se, om vi har fat i et primtal, eller et tal, der kan faktoriseres i mindre primtalsfaktorer.Som hj\303\246lp kan vi bruge kommandoen PrimeFactors fra pakken NumberTheory:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEtUHJpbWVGYWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiMtSSNtbkdGJDYkUSUxODA3RicvRjNRJ25vcm1hbEYnRj4tSSNtb0dGJDYtUSI7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==Ud fra dette output kan vi afl\303\246se to nye primtal til vores liste: 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 og 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139. Indtil videre ser vores liste af primtal s\303\245ledes ud: 2, 3, 7, 13, 43, 139.Det er et ul\303\270st problem i talteori, om man med denne metode i princippet vil kunne konstruere alle primtal.Opgave 11:a) Forts\303\246t beregningen af primtalslisten fra eksempel 4 indtil den indeholder 15 primtal (brug fortsat metoden angivet i Euklids bevis: Find primfaktorerne i (1 + produktet af de forel\303\270big konstruerede primtal), og f\303\270j disse primfaktorer til listen af primtal).b) \303\205ben opgave: Udv\303\246lg selv primtallene 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 og anvend metoden fra beviset for Euklids s\303\246tning til at finde nye primtal. Man kunne f.eks. begynde med listen 3, 11. (Det er hensigtsm\303\246ssigt at stoppe beregningerne, n\303\245r tallene bliver s\303\245 store, at Maple har sv\303\246rt ved at f\303\270lge med).Det er muligt at nuancere metoden til konstruktion af primtal ud fra Euklids bevis s\303\245ledes:F\303\270j kun \303\251n faktor til produktet 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 for hvert trin i algoritmen. Dette uddybes i det f\303\270lgende eksempel:Eksempel 5:I eksempel 4 fandt vi primfaktorerne 13 og 139 til tallet (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGL0YyLUYsNiRRIjdGJ0YvRjItRiw2JFEjNDNGJ0YvLUYzNi1RIitGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGVy1GLDYkUSIxRidGL0Yv).If\303\270lge Euklids algoritme skal vi i n\303\246ste trin finde primfaktorerne i tallet (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), men alternativt kunne vi f\303\270rst finde primfaktorerne i (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GLDYkUSIzRidGL0YyLUYsNiRRIjdGJ0YvRjItRiw2JFEjNDNGJ0YvRjItRiw2JFEjMTNGJ0YvLUYzNi1RIitGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGWi1GLDYkUSIxRidGL0Yv), og vente med at medtage primtallet 139 som faktor: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Men her fandt vi igen to nye primtal, som vi ikke har brugt f\303\270r: Primtallene 53 og 443. I stedet for at bruge 139 som faktor i n\303\246ste trin, kan vi bruge det mindre primtal 53:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEtUHJpbWVGYWN0b3JzRicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JS1GIzYvLUkjbW5HRiQ2JVEiMkYnRjIvRjZRJ25vcm1hbEYnLUkjbW9HRiQ2LlEnJnNkb3Q7RidGMkZBLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlgtRj42JVEiM0YnRjJGQUZDLUY+NiVRIjdGJ0YyRkFGQy1GPjYlUSM0M0YnRjJGQUZDLUY+NiVRIzEzRidGMkZBRkMtRj42JVEjNTNGJ0YyRkEtRkQ2LlEiK0YnRjJGQUZHRkpGTEZORlBGUkZUL0ZXUSwwLjIyMjIyMjJlbUYnL0ZaRmhvLUY+NiVRIjFGJ0YyRkFGMkZBLUZENi5RIjtGJ0YyRkFGRy9GS0YxRkxGTkZQRlJGVEZWL0ZaUSwwLjI3Nzc3NzhlbUYnOpgave 12:I opgave 11a opn\303\245ede vi kun at lave en liste af 15 primtal inden tallene blev s\303\245 store, at Maple skulle bruge meget lang tid til at gennemf\303\270re beregningerne.a) Unders\303\270g hvor mange primtal, der kan konstrueres ved hj\303\246lp af metoden angivet i eksempel 5, f\303\270r Maple giver op.b) \303\205ben opgave: Eksempel 5 tog udgangspunkt i at listen af primtal i starten kun bestod af tallet 2. Gennemf\303\270r metoden i eksempel 5, hvor primtalslisten fra start er forskellig fra 2