{"id":255,"date":"2013-07-05T00:00:26","date_gmt":"2013-07-04T21:00:26","guid":{"rendered":"http:\/\/spektraklet.wordpress.com\/?p=255"},"modified":"2018-03-15T14:58:09","modified_gmt":"2018-03-15T11:58:09","slug":"delbarhet-del-2","status":"publish","type":"post","link":"https:\/\/spektrum.fi\/spektraklet\/delbarhet-del-2\/","title":{"rendered":"Delbarhet, del 2"},"content":{"rendered":"<p>I artikeln <a href=\"http:\/\/spektraklet.wordpress.com\/2013\/04\/02\/matematikhornan-delbarhet\/\">Matematikh\u00f6rnan: Delbarhet<\/a> definierades delbarhetssekvenser f\u00f6r naturliga tal. <em>Delbarhetssekvensen<\/em> f\u00f6r det naturliga talet <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> best\u00e5r av talen <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ei%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^i\\:(mod \\: n)' title='10^i\\:(mod \\: n)' class='latex' \/>, d\u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=i+%5Cin+%5Cmathbb%7BN%7D_%7B0%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='i \\in \\mathbb{N}_{0}' title='i \\in \\mathbb{N}_{0}' class='latex' \/>. I nedanst\u00e5ende tabell ser vi b\u00f6rjan av delbarhetssekvenserna f\u00f6r n\u00e5gra tal:<\/p>\n<ul>\n<li>2: 1, 0, 0, 0, 0, 0&#8230;<\/li>\n<li>3: 1, 1, 1, 1, 1, 1&#8230;<\/li>\n<li>5: 1, 0, 0, 0, 0, 0&#8230;<\/li>\n<li>7: 1, 3, 2, -1, -3, -2&#8230;<\/li>\n<li>9: 1, 1, 1, 1, 1, 1&#8230;<\/li>\n<li>11: 1, -1, 1, -1, 1, -1&#8230;<\/li>\n<\/ul>\n<p>I f\u00f6reg\u00e5ende artikel m\u00e4rkte vi att alla av dessa sekvenser kommer att upprepa sig i n\u00e5got skede. F\u00f6r talen 2 och 5 beror detta p\u00e5 att n\u00e5got av talen \u00e4r noll, och d\u00e4rmed kommer alla d\u00e4rp\u00e5f\u00f6ljande tal att vara noll, ty ifall <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+0+%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 0 \\:(mod \\: n)' title='10^k \\equiv 0 \\:(mod \\: n)' class='latex' \/>, s\u00e5 \u00e4r \u00e4ven <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5E%7Bk%2Bm%7D+%5Cequiv+10%5Ek10%5Em+%5Cequiv+0%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^{k+m} \\equiv 10^k10^m \\equiv 0\\:(mod \\: n)' title='10^{k+m} \\equiv 10^k10^m \\equiv 0\\:(mod \\: n)' class='latex' \/>.<\/p>\n<p>De \u00f6vriga sekvenserna i tabellen inneh\u00e5ller talet 1 p\u00e5 n\u00e5got annat index \u00e4n 0. D\u00e5 upprepas sekvensen, eftersom om <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+0%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 0\\:(mod \\: n)' title='10^k \\equiv 0\\:(mod \\: n)' class='latex' \/>, s\u00e5 g\u00e4ller <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5E%7Bk%2Bm%7D+%5Cequiv+10%5Ek10%5Em+%5Cequiv+10%5Em+%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^{k+m} \\equiv 10^k10^m \\equiv 10^m \\:(mod \\: n)' title='10^{k+m} \\equiv 10^k10^m \\equiv 10^m \\:(mod \\: n)' class='latex' \/>.<\/p>\n<p>Allts\u00e5, ifall en sekvens inneh\u00e5ller talet 0 eller 1 p\u00e5 n\u00e5got annat index \u00e4n 0, kommer sekvensen att upprepas.<\/p>\n<h2>Sekvenser som inneh\u00e5ller talet 0<\/h2>\n<p>F\u00f6r att delbarhetssekvensen f\u00f6r talet <img src='https:\/\/s0.wp.com\/latex.php?latex=n+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n \\in \\mathbb{N}' title='n \\in \\mathbb{N}' class='latex' \/> skall inneh\u00e5lla talet 0, m\u00e5ste det f\u00f6r n\u00e5got <img src='https:\/\/s0.wp.com\/latex.php?latex=k+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k \\in \\mathbb{N}' title='k \\in \\mathbb{N}' class='latex' \/> g\u00e4lla att<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+0+%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 0 \\:(mod \\: n)' title='10^k \\equiv 0 \\:(mod \\: n)' class='latex' \/>, dvs. <img src='https:\/\/s0.wp.com\/latex.php?latex=n+%5Cmid+10%5Ek&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n \\mid 10^k' title='n \\mid 10^k' class='latex' \/>.<\/p>\n<p>Eftersom <img src='https:\/\/s0.wp.com\/latex.php?latex=10+%3D+2%2A5&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10 = 2*5' title='10 = 2*5' class='latex' \/>, kan detta g\u00e4lla endast ifall talet <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> inneh\u00e5ller endast primtalsfaktorerna 2 och 5, dvs. om<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=n%3D2%5Ea5%5Eb&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n=2^a5^b' title='n=2^a5^b' class='latex' \/>, d\u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=a%2Cb+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='a,b \\in \\mathbb{N}' title='a,b \\in \\mathbb{N}' class='latex' \/>.<\/p>\n<p>Ifall <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> kan skrivas i ovanst\u00e5ende form, och vi betecknar <img src='https:\/\/s0.wp.com\/latex.php?latex=k%3Dmax%28%5C%7Ba%2Cb%5C%7D%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k=max(\\{a,b\\})' title='k=max(\\{a,b\\})' class='latex' \/>, g\u00e4ller <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+0%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 0\\:(mod \\: n)' title='10^k \\equiv 0\\:(mod \\: n)' class='latex' \/>.<\/p>\n<h2>Sekvenser som inneh\u00e5ller talet 1<\/h2>\n<p>I vilka fall kan vi d\u00e5 vara s\u00e4kra p\u00e5 att ett tal har en delbarhetssekvens som p\u00e5 n\u00e5got index olika noll har talet 1? Antag att <img src='https:\/\/s0.wp.com\/latex.php?latex=p+%5Cin+%5Cmathbb%7BP%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p \\in \\mathbb{P}' title='p \\in \\mathbb{P}' class='latex' \/> \u00e4r ett primtal. Ifall <img src='https:\/\/s0.wp.com\/latex.php?latex=p&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p' title='p' class='latex' \/> ej \u00e4r talet 2 eller 5, kommer <img src='https:\/\/s0.wp.com\/latex.php?latex=p&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p' title='p' class='latex' \/> inte att dela talet <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k' title='10^k' class='latex' \/> f\u00f6r n\u00e5got <img src='https:\/\/s0.wp.com\/latex.php?latex=k+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k \\in \\mathbb{N}' title='k \\in \\mathbb{N}' class='latex' \/>, allts\u00e5 \u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cnot%5Cequiv+0%5C%3A%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\not\\equiv 0\\:(mod \\: p)' title='10^k \\not\\equiv 0\\:(mod \\: p)' class='latex' \/> f\u00f6r alla <img src='https:\/\/s0.wp.com\/latex.php?latex=k+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k \\in \\mathbb{N}' title='k \\in \\mathbb{N}' class='latex' \/>.<\/p>\n<p>Som bekant bildar modulo-operationen <img src='https:\/\/s0.wp.com\/latex.php?latex=%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='(mod \\: p)' title='(mod \\: p)' class='latex' \/> ringen <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p}' title='\\mathbb{Z}_{p}' class='latex' \/>. Eftersom <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cnot%5Cequiv+0%5C%3A%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\not\\equiv 0\\:(mod \\: p)' title='10^k \\not\\equiv 0\\:(mod \\: p)' class='latex' \/> om p inte \u00e4r primtalet 2 eller 5, g\u00e4ller det att <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cin+%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\in \\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='10^k \\in \\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/>. Dessutom vet vi att om (och endast om) <img src='https:\/\/s0.wp.com\/latex.php?latex=p&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p' title='p' class='latex' \/> \u00e4r ett primtal, \u00e4r ringen <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p}' title='\\mathbb{Z}_{p}' class='latex' \/> en kropp, allts\u00e5 har alla tal f\u00f6rutom noll en multiplikativ invers. Detta \u00e4r ekvivalent med att <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/> bildar en grupp med multiplikationsoperationen.<\/p>\n<p>Delbarhetssekvensen best\u00e5r per definition av talen <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ei%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^i\\:(mod \\: n)' title='10^i\\:(mod \\: n)' class='latex' \/>, d\u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=i+%5Cin+%5Cmathbb%7BN%7D_%7B0%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='i \\in \\mathbb{N}_{0}' title='i \\in \\mathbb{N}_{0}' class='latex' \/>. I fallet d\u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=p&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='p' title='p' class='latex' \/> \u00e4r ett primtal olika 2 och 5, inneh\u00e5lls alla tal i sekvensen allts\u00e5 i den delgrupp av <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/> som talet <img src='https:\/\/s0.wp.com\/latex.php?latex=10&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10' title='10' class='latex' \/> (eller n\u00e4rmare sagt dess ekvivalensklass) genererar. Eftersom gruppen <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/> \u00e4r \u00e4ndlig, kommer \u00e4ven den genererade delgruppen att vara \u00e4ndlig. D\u00e4rmed existerar det ett <img src='https:\/\/s0.wp.com\/latex.php?latex=k+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k \\in \\mathbb{N}' title='k \\in \\mathbb{N}' class='latex' \/> f\u00f6r vilket <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+1+%5C%3A%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 1 \\:(mod \\: p)' title='10^k \\equiv 1 \\:(mod \\: p)' class='latex' \/>.<\/p>\n<p>Talen i sekvensen kommer allts\u00e5 att upprepas efter ett visst index k. N\u00e4rmare sagt, eftersom <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+1+%5Cequiv+10%5E0+%5C%3A%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 1 \\equiv 10^0 \\:(mod \\: p)' title='10^k \\equiv 1 \\equiv 10^0 \\:(mod \\: p)' class='latex' \/>, s\u00e4 g\u00e4ller<\/p>\n<p style=\"text-align:center;\">$latex \\begin{array}{ll}<br \/>\n10^{k+1} \\equiv 10^k10^1 \\equiv 10^1\\\\<br \/>\n10^{k+2} \\equiv 10^k10^2 \\equiv 10^2\\\\<br \/>\n&#8230;\\\\<br \/>\n10^{k+k-1} \\equiv 10^k10^{k-1} \\equiv 10^{k-1}\\\\<br \/>\n\\end{array}$ (mod p)<\/p>\n<p>Sekvensen best\u00e5r allts\u00e5 av cykler med l\u00e4ngden k. Eftersom elementen i cykeln bildar en delgrupp av <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/>, och gruppen <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bp%7D+%5Cchar%60%5C%5C+%5C%7B0%5C%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' title='\\mathbb{Z}_{p} \\char`\\\\ \\{0\\}' class='latex' \/> har p-1 element, vet vi enligt Lagranges sats att cykelns l\u00e4ngd delar talet p-1. Vi ser ocks\u00e5 att eftersom <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5E0+%5Cequiv+1+%5C%3A%28mod+%5C%3A+p%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^0 \\equiv 1 \\:(mod \\: p)' title='10^0 \\equiv 1 \\:(mod \\: p)' class='latex' \/>, kommer cykeln att starta fr\u00e5n det f\u00f6rsta talet (allts\u00e5 talet <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5E0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^0' title='10^0' class='latex' \/>) i sekvensen. Sekvensen har allts\u00e5 ingen s.k. svans f\u00f6re cykeln.<\/p>\n<p>Allts\u00e5 alla primtal utom 2 och 5 har ett index olika noll p\u00e5 vilket delbarhetssekvensen har v\u00e4rdet 1. Naturligtvis kan \u00e4ven icke-primtal ha talet 1 i sin sekvens; i tabellen ovan ser vi att detta g\u00e4ller f\u00f6r talet 9.<\/p>\n<h2>\u00d6vriga fall<\/h2>\n<p>Vi kan vara s\u00e4kra p\u00e5 att delbarhetssekvensen f\u00f6r talet <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> (icke-trivialt) inneh\u00e5ller talet 1 endast om <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u00e4r ett primtal. D\u00e4rmed kunde man f\u00f6rv\u00e4nta sig att det existerar tal vars delbarhetssekvens inneh\u00e5ller varken talet 0 eller talet 1. Det visar sig att detta g\u00e4ller f\u00f6r talet 6. L\u00e5t oss allts\u00e5 unders\u00f6ka dess delbarhetssekvens:<\/p>\n<p>F\u00f6r talet 10 g\u00e4ller det att<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=10+%5Cequiv+4+%5Cequiv+-2+%5C%3A%28mod+%5C%3A+6%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10 \\equiv 4 \\equiv -2 \\:(mod \\: 6)' title='10 \\equiv 4 \\equiv -2 \\:(mod \\: 6)' class='latex' \/>,<\/p>\n<p>och d\u00e4rmed ser vi att<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=10%5E2+%5Cequiv+10%2A10+%5Cequiv+-2%2A%28-2%29+%5Cequiv+4+%5Cequiv+-2+%5C%3A%28mod+%5C%3A+6%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^2 \\equiv 10*10 \\equiv -2*(-2) \\equiv 4 \\equiv -2 \\:(mod \\: 6)' title='10^2 \\equiv 10*10 \\equiv -2*(-2) \\equiv 4 \\equiv -2 \\:(mod \\: 6)' class='latex' \/>.<\/p>\n<p>Av detta f\u00f6ljer naturligtvis direkt att<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek+%5Cequiv+10%5E%7Bk-1%7D+%5Cequiv+%5Ccdots+%5Cequiv+10+%5Cequiv+-2+%5C%3A%28mod+%5C%3A+6%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k \\equiv 10^{k-1} \\equiv \\cdots \\equiv 10 \\equiv -2 \\:(mod \\: 6)' title='10^k \\equiv 10^{k-1} \\equiv \\cdots \\equiv 10 \\equiv -2 \\:(mod \\: 6)' class='latex' \/>.<\/p>\n<p>Allts\u00e5 \u00e4r delbarhetssekvensen f\u00f6r talet 6 1,-2,-2,-2,&#8230; och sekvensen upprepar sig sj\u00e4lv, fast\u00e4n den inte icke-trivialt inneh\u00e5ller talet 0 eller 1.<\/p>\n<p>Vad kan vi d\u00e5 s\u00e4ga om delbarhetssekvensen f\u00f6r talet <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> som ej \u00e4r ett primtal och har andra primtalsfaktorer \u00e4n 2 och 5? Som vi redan tidigare konstaterat, bildar modulo-operationen <img src='https:\/\/s0.wp.com\/latex.php?latex=%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='(mod \\: n)' title='(mod \\: n)' class='latex' \/> ringen <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cmathbb%7BZ%7D_%7Bn%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\mathbb{Z}_{n}' title='\\mathbb{Z}_{n}' class='latex' \/>. Ringen \u00e4r nu inte en kropp, eftersom vi antog att <img src='https:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> inte \u00e4r ett primtal. Vi vet dock \u00e4nd\u00e5 att ifall vi multiplicerar tv\u00e5 tal i ringen, kommer resultatet \u00e4ven att befinna sig i ringen.<\/p>\n<p>Vi har allts\u00e5 endast n stycken olika m\u00f6jliga multiplikationsresultat. D\u00e4rmed om vi ber\u00e4knar talen <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Em+%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^m \\:(mod \\: n)' title='10^m \\:(mod \\: n)' class='latex' \/>, d\u00e4r <img src='https:\/\/s0.wp.com\/latex.php?latex=m+%5Cin+%5Cmathbb%7BN%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m \\in \\mathbb{N}' title='m \\in \\mathbb{N}' class='latex' \/>, m\u00e5ste vi i n\u00e5got skede st\u00f6ta p\u00e5 ett tal som f\u00f6rekommit tidigare i sekvensen. Vi vet \u00e4ven att detta h\u00e4nder senast, d\u00e5 <img src='https:\/\/s0.wp.com\/latex.php?latex=m%3Dn&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m=n' title='m=n' class='latex' \/>, eftersom vi inte kan ha <img src='https:\/\/s0.wp.com\/latex.php?latex=n%2B1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n+1' title='n+1' class='latex' \/> olika tal i ringen. Allts\u00e5, vi hittar ett tal <img src='https:\/\/s0.wp.com\/latex.php?latex=m&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m' title='m' class='latex' \/> och ett annat tal <img src='https:\/\/s0.wp.com\/latex.php?latex=k+%3C+m&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k &lt; m' title='k &lt; m' class='latex' \/> f\u00f6r vilka<\/p>\n<p style=\"text-align:center;\"><img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Em+%5Cequiv+10%5Ek+%5C%3A%28mod+%5C%3A+n%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^m \\equiv 10^k \\:(mod \\: n)' title='10^m \\equiv 10^k \\:(mod \\: n)' class='latex' \/>.<\/p>\n<p>Av detta f\u00f6ljer naturligtvis att<\/p>\n<p style=\"text-align:center;\">$latex \\begin{array}{ll}<br \/>\n10^{m+1} \\equiv 10^m10^1 \\equiv 10^k10^1 \\equiv 10^{k+1}\\\\<br \/>\n10^{m+2} \\equiv 10^m10^2 \\equiv 10^k10^2 \\equiv 10^{k+2}\\\\<br \/>\n&#8230;\\\\<br \/>\n10^{m+(m-k)-1} \\equiv 10^{k+(m-k)-1} \\equiv 10^{m-1}\\\\<br \/>\n10^{m+(m-k)} \\equiv 10^{k+(m-k)} \\equiv 10^k\\\\<br \/>\n\\end{array}$ (mod n),<\/p>\n<p>allts\u00e5 g\u00e5r sekvensen efter talet <img src='https:\/\/s0.wp.com\/latex.php?latex=10%5Ek&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='10^k' title='10^k' class='latex' \/> in i en cykel med l\u00e4ngden <img src='https:\/\/s0.wp.com\/latex.php?latex=m-k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='m-k' title='m-k' class='latex' \/>. Till skillnad fr\u00e5n sekvenserna f\u00f6r primtal, kan denna sekvens ha en s.k. svans i b\u00f6rjan f\u00f6rr\u00e4n cykeln b\u00f6rjar. Detta beror p\u00e5 att talet 1 inte n\u00f6dv\u00e4ndigtvis f\u00f6rekommer i sekvensen f\u00f6rutom i b\u00f6rjan.<\/p>\n<h2>\u00d6ppna fr\u00e5gor<\/h2>\n<p>Talen 3 och 9 har exakt likadan delbarhetssekvens; den best\u00e5r endast av ettor. Finns det andra talpar som har exakt likadan delbarhetssekvens? Om det finns, kan vi hitta n\u00e5got samband f\u00f6r tal som har likadan delbarhetssekvens?<\/p>\n<p>Finns det n\u00e5got samband mellan primtalsutvecklingen och delbarhetssekvensen f\u00f6r tal?<\/p>\n<p>Kan vi p\u00e5 n\u00e5got vettigt sett gruppera icke-primtal i olika grupper, t.ex. enligt l\u00e4ngden p\u00e5 cykeln?<\/p>\n<h3>Litteratur<\/h3>\n<p>Allm\u00e4n algebra:<\/p>\n<p>Jokke H\u00e4s\u00e4, Johanna R\u00e4m\u00f6: Johdatus abstraktiin algebraan, Gaudeamus, 2012.<br \/>\nTauno Mets\u00e4nkyl\u00e4, Marjatta N\u00e4\u00e4t\u00e4nen: Algebra, Limes, 2005.<\/p>\n<p>Nyttiga wikipedia-l\u00e4nkar:<\/p>\n<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Group_%28mathematics%29\" target=\"_blank\">Group<\/a><br \/>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Ring_%28mathematics%29\" target=\"_blank\">Ring<\/a><br \/>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Finite_field\" target=\"_blank\">Finite field (\u00e4ndlig kropp)<\/a><\/p>\n<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Lagrange%27s_theorem_%28group_theory%29\" target=\"_blank\">Lagrange&#8217;s theorem<\/a><br \/>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Fundamental_theorem_of_arithmetic\" target=\"_blank\">Fundamental theorem of arithmetic<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I artikeln Matematikh\u00f6rnan: Delbarhet definierades delbarhetssekvenser f\u00f6r naturliga tal. Delbarhetssekvensen f\u00f6r det naturliga talet best\u00e5r av talen , d\u00e4r . I nedanst\u00e5ende tabell ser vi b\u00f6rjan av delbarhetssekvenserna f\u00f6r n\u00e5gra tal: 2: 1, 0, 0, 0, 0, 0&#8230; 3: 1, 1, 1, 1, 1, 1&#8230; 5: 1, 0, 0, 0, 0, 0&#8230; 7: 1, 3, &hellip; <a href=\"https:\/\/spektrum.fi\/spektraklet\/delbarhet-del-2\/\" class=\"more-link\">Forts\u00e4tt l\u00e4sa <span class=\"screen-reader-text\">Delbarhet, del 2<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82,5],"tags":[57],"_links":{"self":[{"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/255"}],"collection":[{"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/comments?post=255"}],"version-history":[{"count":1,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/255\/revisions"}],"predecessor-version":[{"id":1943,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/255\/revisions\/1943"}],"wp:attachment":[{"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/media?parent=255"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/categories?post=255"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/tags?post=255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}