{"id":8920,"date":"2016-03-12T18:23:08","date_gmt":"2016-03-12T10:23:08","guid":{"rendered":"http:\/\/stuif.com\/blog\/?p=8920"},"modified":"2016-03-12T23:07:10","modified_gmt":"2016-03-12T15:07:10","slug":"the-largest-prime-number","status":"publish","type":"post","link":"https:\/\/stuif.com\/blog\/?p=8920","title":{"rendered":"The Largest Prime Number"},"content":{"rendered":"<p>On 7 January 2016 a &#8220;new&#8221;\u00a0<a href=\"http:\/\/www.theguardian.com\/science\/alexs-adventures-in-numberland\/2016\/jan\/19\/largest-prime-number-yet-discovered-has-22-million-digits\">large prime number was discovered<\/a>,\u00a0with more than <span style=\"color: #00ffff;\">22 million digits<\/span>. Time for a blog about these numbers, which\u00a0have fascinated mathematicians from Greek antiquity until present times.<\/p>\n<p>Prime numbers are numbers that can only be divided by 1 and itself. For example 7 is a\u00a0prime number, but 6 is not because it can be divided by 2 and 3. . Here is a list of the 168 prime numbers smaller than 1000.\u00a0The number 2 is the only even prime, all others are odd.<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/prime-numbers1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9127\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9127\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/prime-numbers1.jpg\" data-orig-size=\"588,393\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"prime-numbers\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/prime-numbers1-300x201.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/prime-numbers1.jpg\" tabindex=\"0\" role=\"button\" class=\"alignnone  wp-image-9127\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/prime-numbers1.jpg\" alt=\"prime-numbers\" width=\"439\" height=\"298\" \/><\/a><\/p>\n<p>How many prime numbers are there?<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euclides1.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9155\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9155\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euclides1.jpg\" data-orig-size=\"425,300\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Euclides\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euclides1-300x212.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euclides1.jpg\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9155 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euclides1.jpg\" alt=\"Euclides\" width=\"177\" height=\"128\" \/><\/a><\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Euclid\">Euclides<\/a>, the famous Greek mathematician, living in present-day Egypt \u00a0around 300 BC, already proved that\u00a0their number is infinite, and his proof is so elementary, that I often\u00a0presented it to my students when I was a teacher, as an example of what is called <a href=\"https:\/\/en.wikipedia.org\/wiki\/Reductio_ad_absurdum\">Reductio ad Absurdum<\/a>.<\/p>\n<p>Euclides&#8217; proof: \u00a0Assume that you have a <span style=\"color: #00ffff;\">complete<\/span> list of all prime numbers.<\/p>\n<ul>\n<li>Multiply them together and add 1. Call this number X.<\/li>\n<li>Because of the added 1, this number X can not be divided by any prime number in your list\u00a0(there will always be a reminder 1)!<\/li>\n<li>So there are only two possibilities, either\u00a0X is prime itself, or it can be divided by a prime number outside your list. In both cases it shows\u00a0your list was <span style=\"color: #00ffff;\">incomplete<\/span>.<\/li>\n<li>Therefore our assumption was wrong and the list of prime numbers is infinite!<\/li>\n<\/ul>\n<p>How to find out if a number X is prime? \u00a0 Do we\u00a0 have to check whether X\u00a0is divisible by any number, smaller than X ? That would be a tedious job. Fortunately it is not as bad as that&#8230;:-). Because it\u00a0is easy to see that we only have to check whether\u00a0X is divisible by any\u00a0<span style=\"color: #00ffff;\">prime number, smaller than the square root of X<\/span>.<\/p>\n<p>For example X=283, is it prime? The square root of 283 = 16.82&#8230;, so we have only to check division by 2,3,5,7,11 and 13.<\/p>\n<ul>\n<li>283 \/ 2 = 141 rest 1<\/li>\n<li>283 \/ 3 = 94 rest 1<\/li>\n<li>283 \/ 5 = 56 rest 3<\/li>\n<li>283 \/ 7 = 40 rest 3<\/li>\n<li>283 \/ 11 = 25 rest 8<\/li>\n<li>283 \/ 13 = 21 rest 10<\/li>\n<\/ul>\n<p>So 283 is a prime number!<\/p>\n<p>This procedure is called <a href=\"https:\/\/en.wikipedia.org\/wiki\/Trial_division\">Trial Division<\/a>. For large numbers it becomes time consuming. For example, we want to check\u00a0if\u00a01000.003\u00a0is prime.\u00a0\u00a0There are 168 prime numbers smaller than 1000, so we have to do 168 divisions to finally conclude that, yes, 1000.003 is prime. Repeating this procedure for 999.997, you will\u00a0find that this number is not prime,\u00a0it can be divided by 757.<\/p>\n<p><span style=\"color: #00ffff;\">Imagine that you have to do these divisions with only pen and paper!<\/span><\/p>\n<p>Back to the recently discovered large mega-prime. It is a so-called Mersenne prime, one less than a power of 2: \u00a0<span style=\"color: #00ffff;\">M<sub>p<\/sub> = 2<sup>p<\/sup> \u2212 1\u00a0<\/span>with p itself a\u00a0prime number.<\/p>\n<ul>\n<li><span style=\"color: #00ffff;\">M<sub>2<\/sub><\/span> = 2<sup>2<\/sup> \u2212 1 = 4 &#8211; 1 = 3 prime!<\/li>\n<li><span style=\"color: #00ffff;\">M<sub>3<\/sub><\/span> = 2<sup>3<\/sup> \u2212 1 = 8 &#8211; 1 = 7 prime!<\/li>\n<li><span style=\"color: #00ffff;\">M<sub>5<\/sub><\/span> = 2<sup>5<\/sup> \u2212 1 = 32 &#8211; 1 = 31 prime!<\/li>\n<li><span style=\"color: #00ffff;\">M<sub>7<\/sub><\/span> = 2<sup>7<\/sup> \u2212 1 = 128 &#8211; 1 = 127 prime!<\/li>\n<\/ul>\n<p><span style=\"color: #00ffff;\">Could this be a rule to create prime numbers?<\/span> Unfortunately that is not the case.\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0<span style=\"color: #00ffff;\">M<sub>11<\/sub><\/span> = 2<sup>11<\/sup> \u2212 1 = 2048 &#8211; 1 = 2047 = 23 * 89 , not prime!<br \/>\nHowever the next one <span style=\"color: #00ffff;\">M<sub>13<\/sub><\/span> = 2<sup>13<\/sup> \u2212 1 = 8192 &#8211; 1 = 8191 is\u00a0again prime.<br \/>\nAs are <span style=\"color: #00ffff;\">M<sub>17<\/sub><\/span> = 131.071 and <span style=\"color: #00ffff;\">M<sub>19<\/sub><\/span> = 524.287. The last two are already quite large, in 1588 the Italian mathematician Cataldi had proven\u00a0by trial division that they were prime.<\/p>\n<p>Why are these numbers called Mersenne primes?<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Marin_mersenne.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9160\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9160\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Marin_mersenne.jpg\" data-orig-size=\"450,571\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Marin_mersenne\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Marin_mersenne-236x300.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Marin_mersenne.jpg\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9160 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Marin_mersenne.jpg\" alt=\"Marin_mersenne\" width=\"153\" height=\"191\" \/><\/a><\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Marin_Mersenne\">Marin Mersenne<\/a> was a French priest with an interest in mathematics, theology and philosophy.<\/p>\n<p>He published in 1644 a list of these numbers 2<sup>p<\/sup> \u2212 1, stating that they were prime for p \u00a0= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and not for any other p\u00a0below\u00a0257.<\/p>\n<p>His list was <span style=\"color: #00ffff;\">incomplete<\/span> and <span style=\"color: #00ffff;\">incorrect<\/span>, but still these prime numbers carry his name&#8230;:-)<\/p>\n<p><span style=\"color: #00ffff;\">Incorrect<\/span>, because\u00a0<span style=\"color: #00ffff;\">M<sub>67<\/sub><\/span> and <span style=\"color: #00ffff;\">M<sub>257<\/sub><\/span> are composite<br \/>\n<span style=\"color: #00ffff;\">Incomplete<\/span>, because <span style=\"color: #00ffff;\">M<sub>61<\/sub><\/span> , <span style=\"color: #00ffff;\">M<sub>89<\/sub><\/span> and <span style=\"color: #00ffff;\">M<sub>107<\/sub><\/span> are prime<\/p>\n<p>Mersenne was correct that <span style=\"color: #00ffff;\">M<sub>31<\/sub><\/span> = 2.147.483.647 is prime, but how could he know? This is a big number, the square root is ~ 46.340, so he should first determine all prime numbers smaller\u00a0than 46.340 \u00a0(there are 4792) and then perform trial division for all those 4792 numbers.\u00a0It must have been a lucky guess. And certainly it was a guess for\u00a0<span style=\"color: #00ffff;\">M<sub>127<\/sub><\/span>\u00a0=\u00a0170.141.183.460.469.231.731.687.303.715.884.105.727 \ud83d\ude42<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euler.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9153\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9153\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euler.jpg\" data-orig-size=\"460,597\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;Photograph: Kunstmuseum Basel\\\/Wi&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;18th century mathematician Leonhard Euler&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"18th century mathematician Leonhard Euler\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euler-231x300.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euler.jpg\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9153 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Euler.jpg\" alt=\"18th century mathematician Leonhard Euler\" width=\"173\" height=\"221\" \/><\/a><\/p>\n<p>It was only in 1772, more than a century later, \u00a0that the great mathematician <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leonhard_Euler\">Leonhard Euler<\/a> proved\u00a0the primality of\u00a0<span style=\"color: #00ffff;\">M<sub>31<\/sub><\/span>.<\/p>\n<p>By a clever analysis of the general structure of Mersenne numbers, he managed to reduce the number of trial divisions to 84 !<\/p>\n<p>Still a big job (pen and paper), the story is that he had a team of helpers to do the actual calculations.<\/p>\n<p>This was the last\u00a0result using\u00a0trial division. For more than a\u00a0century no developments regarding Mersenne primes\u00a0took place.<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Elucas_1.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9168\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9168\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Elucas_1.png\" data-orig-size=\"315,503\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Elucas_1\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Elucas_1-188x300.png\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Elucas_1.png\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9168 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/Elucas_1.png\" alt=\"Elucas_1\" width=\"119\" height=\"184\" \/><\/a><\/p>\n<p>Until 1857, when<a href=\"https:\/\/en.wikipedia.org\/wiki\/%C3%89douard_Lucas\"> \u00c9douard Lucas<\/a>, a young French boy (15 years old), gets interested to prove\u00a0that\u00a0<span style=\"color: #00ffff;\">M<sub>127<\/sub><\/span> is prime.<\/p>\n<p>As trial division is not feasible for these large numbers, he studies the structure of the Mersenne numbers and develops a method to check the primality without trial divisions.<\/p>\n<p>After 19 (!) years of testing his methods, he is convinced and announces in 1876 that <span style=\"color: #00ffff;\">M<sub>127<\/sub><\/span> is prime. The 9th Mersenne prime!<\/p>\n<p>His approach, later refined by others,\u00a0is still used in the search for new Mersenne primes. Characteristic for this\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Lucas%E2%80%93Lehmer_primality_test\">Lucas\u2013Lehmer primality test<\/a>\u00a0is that it can decide that a Mersenne number is NOT prime, without finding the factors of this number. For example, using this test, we find that\u00a0<span style=\"color: #00ffff;\">M<sub>257<\/sub><\/span> = 231.584.178.474.632.390.847.141.970.017.375.815.706.539.969.331.281.128.078.915.168.015.826.259.279.871 is composite, but we don&#8217;t know its factors&#8230;:-)<\/p>\n<p>With\u00a0Lucas&#8217; method, in the following years\/decades the\u00a0primality of\u00a0<span style=\"color: #00ffff;\">M<sub>61<\/sub><\/span> , <span style=\"color: #00ffff;\">M<sub>89<\/sub><\/span> and <span style=\"color: #00ffff;\">M<sub>107<\/sub><\/span> is proven. Still using pen and paper!<\/p>\n<p>New activity starts only in the 20th century when the first computers are built.<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/SWAC_001.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9172\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9172\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/SWAC_001.jpg\" data-orig-size=\"1280,993\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"SWAC_001\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/SWAC_001-300x233.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/SWAC_001-1024x794.jpg\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9172 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/SWAC_001.jpg\" alt=\"SWAC_001\" width=\"251\" height=\"197\" \/><\/a><\/p>\n<p>One of them is the famous\u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/SWAC_(computer)\">SWAC<\/a> computer, built in 1950. Nowadays a PC or even a tablet would be more powerful.<\/p>\n<p>In 1952 it was\u00a0used to check new Mersenne primes. Within one year 5 new ones were\u00a0found, for p = 521, 607, 1279, 2203 \u00a0and 2281.\u00a0Still using the methods developed by Lucas<\/p>\n<p>Very large numbers! Here is\u00a0prime number <span style=\"color: #00ffff;\">M<sub>2281<\/sub><\/span> with 687 digits.\u00a0To make it more readable, spaces have been inserted after three digits.<\/p>\n<p>446 087 557 183 758 429 571 151 706 402 101 809 886 208 632 412 859 901 111 991 219 963 404 685 792 820 473 369 112 545 269 003 989 026 153 245 931 124 316 702 395 758 705 693 679 364 790 903 497 461 147 071 065 254 193 353 938 124 978 226 307 947 312 410 798 874 869 040 070 279 328 428 810 311 754 844 108 094 878 252 494 866 760 969 586 998 128 982 645 877 596 028 979 171 536 962 503 068 429 617 331 702 184 750 324 583 009 171 832 104 916 050 157 628 886 606 372 145 501 702 225 925 125 224 076 829 605 427 173 573 964 812 995 250 569 412 480 720 738 476 855 293 681 666 712 844 831 190 877 620 606 786 663 862 190 240 118 570 736 831 901 886 479 225 810 414 714 078 935 386 562 497 968 178 729 127 629 594 924 411 960 961 386 713 946 279 899 275 006 954 917 139 758 796 061 223 803 393 537 381 034 666 494 402 951 052 059 047 968 693 255 388 647 930 440 925 104 186 817 009 640 171 764 133 172 418 132 836 351<\/p>\n<p><a href=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/IBM_7090_computer.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"9186\" data-permalink=\"https:\/\/stuif.com\/blog\/?attachment_id=9186\" data-orig-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/IBM_7090_computer.jpg\" data-orig-size=\"631,489\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"IBM_7090_computer\" data-image-description=\"\" data-image-caption=\"\" data-medium-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/IBM_7090_computer-300x232.jpg\" data-large-file=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/IBM_7090_computer.jpg\" tabindex=\"0\" role=\"button\" class=\"  wp-image-9186 alignleft\" src=\"https:\/\/stuif.com\/blog\/wp-content\/uploads\/2016\/03\/IBM_7090_computer.jpg\" alt=\"IBM_7090_computer\" width=\"257\" height=\"202\" \/><\/a><\/p>\n<p>Computers became more powerful, and new Mersenne primes were\u00a0discovered. In 1961 the first Mersenne prime with more than 1000 digits was\u00a0found, <span style=\"color: #00ffff;\">M<sub>4253<\/sub><\/span>, using an IBM 7090 mainframe computer (pic left)<\/p>\n<p>And in 1979 the first Mersenne prime with more than 10.000 digits was\u00a0found, <span style=\"color: #00ffff;\">M<sub>44.497<\/sub><\/span>, using a Cray supercomputer.<\/p>\n<p>You might expect that the recently discovered Mersenne prime\u00a0<span style=\"color: #00ffff;\">M<sub>74.207.281<\/sub><\/span> with more than 22 million digits has been found using a super-super computer&#8230;:-) \u00a0But that is\u00a0not the case!\u00a0Actually PC&#8217;s were used, not one but many,\u00a0working together!<\/p>\n<p>In 1996 the Great Internet Mersenne Prime Search ( <a href=\"https:\/\/en.wikipedia.org\/wiki\/Great_Internet_Mersenne_Prime_Search\">GIMPS<\/a>) project was started. It is an example of what is called \u00a0<a href=\"https:\/\/en.wikipedia.org\/wiki\/Distributed_computing\">distributed computing<\/a>.\u00a0\u00a0A PC will often be idle, so why not \u00a0let it work during that time\u00a0for a project such as GIMPS. Just download some software and your PC will try to find a new Mersenne Prime. Many thousands of volunteers are doing this. And with success<\/p>\n<p>Since 1996, 15 new Mersenne primes have been found, all of them using GIMPS!<\/p>\n<p>Of course\u00a0finding a new Mersenne prime has no scientific value, it is just an intellectual challenge. But\u00a0you might win a prize!<\/p>\n<p>When\u00a0in 1999 the first Mersenne prime was found with more than 1 million digits,<span style=\"color: #00ffff;\">M<sub>6.972.593<\/sub><\/span>\u00a0, the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Electronic_Frontier_Foundation\">Electronic Frontier Foundation<\/a> awarded this result with a prize of 50.000 US$. In 2008 <span style=\"color: #00ffff;\">M<sub>37.156.667<\/sub><\/span> was found, with more than 10 million digits. The award was 100.000 US$<\/p>\n<p>Two more prizes have not yet been awarded<\/p>\n<ul>\n<li>150.000 US$ to the first individual or group who discovers a prime number with at least 100 million digits<\/li>\n<li>250.000 US$ to the first individual or group who discovers a prime number with at least \u00a01 billion digits<\/li>\n<\/ul>\n<p>The latest Mersenne prime <span style=\"color: #00ffff;\">M<sub>74.207.281<\/sub><\/span> has 22.338.618 digits, not yet enough for the next reward<\/p>\n<p>So, why not join GIMPS! \u00a0I did&#8230;.:-)<\/p>\n<p>For this blog I have made extensive use of <a href=\"https:\/\/primes.utm.edu\/\">The Prime Pages<\/a><\/p>\n<p>Large numbers have been calculated using the <a href=\"https:\/\/defuse.ca\/big-number-calculator.htm\">Online Big Number Calculator<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>On 7 January 2016 a &#8220;new&#8221;\u00a0large prime number was discovered,\u00a0with more than 22 million digits. Time for a blog about these numbers, which\u00a0have fascinated mathematicians from Greek antiquity until present times. Prime numbers are numbers that can only be divided &hellip; <a href=\"https:\/\/stuif.com\/blog\/?p=8920\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[32],"tags":[],"class_list":["post-8920","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2LqIR-2jS","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/8920","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8920"}],"version-history":[{"count":103,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/8920\/revisions"}],"predecessor-version":[{"id":9218,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/8920\/revisions\/9218"}],"wp:attachment":[{"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/stuif.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}