{"id":26672,"date":"2022-06-15T16:46:37","date_gmt":"2022-06-15T14:46:37","guid":{"rendered":"https:\/\/mahifi.se\/?page_id=26672"},"modified":"2023-02-18T15:55:15","modified_gmt":"2023-02-18T13:55:15","slug":"primtal-pa-formen","status":"publish","type":"page","link":"https:\/\/mahifi.se\/?page_id=26672","title":{"rendered":"Primtal p\u00e5 formen"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"26672\" class=\"elementor elementor-26672\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-78ea706 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"78ea706\" data-element_type=\"section\" data-e-type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a9c193a\" data-id=\"a9c193a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7ea6880 elementor-widget elementor-widget-heading\" data-id=\"7ea6880\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Hitta primtal p\u00e5 formen <span class=\"katex-eq\" data-katex-display=\"false\"> 2^n-1<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-df05a5d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"df05a5d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1b8568e\" data-id=\"1b8568e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0035fbb elementor-widget elementor-widget-text-editor\" data-id=\"0035fbb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><b>V\u00e5ra f\u00f6rsta och st\u00f6rsta primtal<\/b><\/p>\n<p>Primtal \u00e4r n\u00e5got som m\u00e4nniskan har varit medveten om mycket l\u00e4nge. En primtal definieras som ett tal som \u00e4r st\u00f6rre \u00e4n 1 och enbart \u00e4r delbart med sig sj\u00e4lvt och 1. Talet g\u00e5r allts\u00e5 inte att dela upp i faktorer av n\u00e5gra andra tal. Primtal \u00e4r fundamentala f\u00f6r matematiken eftersom alla naturliga tal &gt;1 antingen \u00e4r primtal eller faktorer av primtal, enligt aritmetikens fundamentalsats.&nbsp;<\/p>\n<p>Vi vet ocks\u00e5 att det finns o\u00e4ndligt m\u00e5nga primtal efter ett bevis som finns i Euklides Elementa som formulerades 300 f.kr. (du kan f\u00f6rs\u00f6ka bevisa det med lite v\u00e4gledning i <a href=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Roliga-artitmetiska-uppgifter.pdf\">f\u00f6ljande dokument<\/a>). I och med att det finns o\u00e4ndligt m\u00e5nga primtal har m\u00e4nniskan alltid varit intresserad av vilken som \u00e4r det st\u00f6rsta m\u00f6jliga primtal som vi m\u00e4nniskor kan hitta. Metoderna har varit olika genom historien. Tidigt unders\u00f6kte man primtal genom att testa vilka delare ett tal hade. Det var en tidskr\u00e4vande aktivitet men gav s\u00e4kra resultat. Fram till 1400-talet hade man inte hittat s\u00e4rskilt m\u00e5nga primtal men 1456 hittade en anonym matematiker primtalet 8191. Ungef\u00e4r 100 \u00e5r senare kunde den franska matematikern <b>Martin Mersenne<\/b> (1588-1648) visa p\u00e5 att primtal kunde skrivas p\u00e5 formen <span class=\"katex-eq\" data-katex-display=\"false\"> 2^n-1 <\/span>. Primtal som g\u00e5r att skrivas p\u00e5 den formen kallas idag f\u00f6r Mersenneprimtal. En matematiker som var samtida med Mersenne var <b>Cataldi <\/b>(1548-1636)<b>&nbsp;<\/b>som hittade tv\u00e5 primtal som kunde skrivas p\u00e5 formen&nbsp;<span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2^n-1 <\/span> n\u00e4mligen&nbsp;<\/span><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2^{17}-1=131071 <\/span> och&nbsp;<\/span><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2^{19}-1=524287 <\/span>. <b>Leonhard Euler<\/b> kunde senare visa att&nbsp;<\/span><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 2^{31}-1=2147483647 <\/span><\/span><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\">&nbsp;var ett primtal.&nbsp;<\/span><\/p>\n<p><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem;\">Progressionen fortsatte under 100 \u00e5r men stannade till slut n\u00e4r den franska matematikern Edouard Lucas efter 19 \u00e5r av unders\u00f6kning lyckades visa att <span class=\"katex-eq\" data-katex-display=\"false\"> 2^{127}-1=170141183460469231731731687303715884105727 <\/span><\/span><span style=\"background-color: var(--ast-global-color-4); color: var(--ast-global-color-3); font-size: 1rem; font-style: inherit; font-weight: inherit;\">&nbsp;var ett primtal. Idag \u00e4r det primtalet fortfarande det st\u00f6rsta som m\u00e4nniskan tagit fram f\u00f6r hand.&nbsp;<\/span><\/p>\n<p><span style=\"font-size: 1rem; font-style: inherit; font-weight: inherit; color: var(--ast-global-color-3); background-color: var(--ast-global-color-4);\">Under 1950-talet hade m\u00e4nniskan b\u00f6rjat producera maskiner som kunde utf\u00f6ra matematiska ber\u00e4kningar i en takt som den m\u00e4nskliga hj\u00e4rnan inte kunde. Det resulterade ocks\u00e5 i att man kunde unders\u00f6ka stora tal om de var primtal mycket mer effektivt. Man utnyttjade faktumet att man visste att flera primtal kunde g\u00e5 att skrivas p\u00e5 formen <span class=\"katex-eq\" data-katex-display=\"false\"> 2^n-1<\/span>. Under 1952 hittade man 4 nya Mersenneprimtal det st\u00f6rsta var <span class=\"katex-eq\" data-katex-display=\"false\"> 2^{2281}-1<\/span> som jag inte kommer skriva ut eftersom talet inneh\u00e5ller mer \u00e4n 687 siffror.&nbsp;<\/span><\/p>\n<p>I och med att datorerna har blivit b\u00e4ttre har ocks\u00e5 datorernas f\u00f6rm\u00e5ga att hitta stora primtal f\u00f6rb\u00e4ttrats. Mellan 1957 &#8211; 2018 har man hittat 33 nya Mersenneprimtal d\u00e4r det st\u00f6rsta primtalet \u00e4r <span class=\"katex-eq\" data-katex-display=\"false\"> 2^{82589933}-1 <\/span>, primtalet inneh\u00e5ller 24862048 siffror. T\u00e4nk vad fantastiskt att ett tal som \u00e4r s\u00e5 stort fortfarande bara kan dela sig sj\u00e4lv och 1. Under de senaste \u00e5ren har ett projekt som heter GIMPS varit aktivt som enbart har som uppgift att hitta stora primtal. Man anv\u00e4nder de mest avancerade datorerna f\u00f6r att m\u00f6jligg\u00f6ra ber\u00e4kningarna som kr\u00e4vs f\u00f6r att unders\u00f6ka om talen \u00e4r primtal. Det tar ocks\u00e5 s\u00e5klart l\u00e4ngre och l\u00e4ngre tid desto st\u00f6rre talen blir. Samtidigt vi vet som sagt att det finns o\u00e4ndligt m\u00e5nga d\u00e4rf\u00f6r kommer vi med r\u00e4tt metoder alltid hitta nya primtal.&nbsp;<\/p>\n<p><span style=\"background-color: var(--ast-global-color-4);\">&nbsp;<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-2b83edd\" data-id=\"2b83edd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d337914 elementor-widget elementor-widget-image\" data-id=\"d337914\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"304\" height=\"372\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Euklides.jpg\" class=\"attachment-large size-large wp-image-26178\" alt=\"\" srcset=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Euklides.jpg 304w, https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Euklides-245x300.jpg 245w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d8186a5 elementor-widget elementor-widget-text-editor\" data-id=\"d8186a5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: center;\">Euklides &#8211; F\u00f6rsta mannen som (eventuellt) bevisade att det finns o\u00e4ndligt med primtal&nbsp;<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0f08d39 elementor-widget elementor-widget-image\" data-id=\"0f08d39\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"450\" height=\"571\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Marin_mersenne.jpeg\" class=\"attachment-large size-large wp-image-26690\" alt=\"\" srcset=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Marin_mersenne.jpeg 450w, https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Marin_mersenne-236x300.jpeg 236w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d867cae elementor-widget elementor-widget-text-editor\" data-id=\"d867cae\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: center;\">Martin Mersenne\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f6dc8c9 elementor-widget elementor-widget-image\" data-id=\"f6dc8c9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"432\" height=\"326\" src=\"https:\/\/mahifi.se\/wp-content\/uploads\/2022\/06\/Digits_in_largest_prime_found_as_a_function_of_time.svg\" class=\"attachment-large size-large wp-image-26704\" alt=\"\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a32f537 elementor-widget elementor-widget-text-editor\" data-id=\"a32f537\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: center;\">Graf som visar p\u00e5 n\u00e4r man fann primtal och dess storlek. Man kan konstatera att de hittar f\u00e4rre desto st\u00f6rre primtalen blir.\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Hitta primtal p\u00e5 formen V\u00e5ra f\u00f6rsta och st\u00f6rsta primtal Primtal \u00e4r n\u00e5got som m\u00e4nniskan har varit medveten om mycket l\u00e4nge. En primtal definieras som ett tal som \u00e4r st\u00f6rre \u00e4n 1 och enbart \u00e4r delbart med sig sj\u00e4lvt och 1. Talet g\u00e5r allts\u00e5 inte att dela upp i faktorer av n\u00e5gra andra tal. Primtal \u00e4r [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"class_list":["post-26672","page","type-page","status-publish","hentry"],"aioseo_notices":[],"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"joji9555@hotmail.com","author_link":"https:\/\/mahifi.se\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"Hitta primtal p\u00e5 formen V\u00e5ra f\u00f6rsta och st\u00f6rsta primtal Primtal \u00e4r n\u00e5got som m\u00e4nniskan har varit medveten om mycket l\u00e4nge. En primtal definieras som ett tal som \u00e4r st\u00f6rre \u00e4n 1 och enbart \u00e4r delbart med sig sj\u00e4lvt och 1. Talet g\u00e5r allts\u00e5 inte att dela upp i faktorer av n\u00e5gra andra tal. Primtal \u00e4r&hellip;","_links":{"self":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26672","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mahifi.se\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26672"}],"version-history":[{"count":46,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26672\/revisions"}],"predecessor-version":[{"id":29888,"href":"https:\/\/mahifi.se\/index.php?rest_route=\/wp\/v2\/pages\/26672\/revisions\/29888"}],"wp:attachment":[{"href":"https:\/\/mahifi.se\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26672"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}