{"id":2714,"date":"2009-06-22T08:15:02","date_gmt":"2009-06-22T00:15:02","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=2714"},"modified":"2014-07-06T10:49:52","modified_gmt":"2014-07-06T02:49:52","slug":"largest-known-prime-243112609-1","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/largest-known-prime-243112609-1-2714","title":{"rendered":"Largest known prime: 2^43,112,609 - 1"},"content":{"rendered":"

A reader wrote to say that there was an error on the page Number Properties<\/a> in Interactive Mathematics. It said:<\/p>\n

\n

213466917<\/sup> − 1. (This is huge - around 4 \u00d7 10506,756<\/sup>). <\/p>\n<\/blockquote>\n

When I wrote the page, this was indeed the highest known prime, but in 2008, the GIMPS project (the distributed computing Great Internet Mersenne Prime Search) found an even larger one:<\/p>\n

243,112,609<\/sup> − 1<\/p>\n

This prime has around 13 million<\/b> digits. Here are some of them (with millions of digits omitted in the middle):<\/p>\n

\"largest-prime\"
\n[Image source: ScienceNews - no longer available]<\/p>\n

It's an example of a Mersenne Prime, which are of the form 2x<\/i><\/sup> − 1.<\/p>\n

The first Mersenne Prime is 22<\/sup> − 1 = 3, and the next is 23<\/sup> − 1 = 7. <\/p>\n

But not all numbers in this form are prime, for example 24<\/sup> − 1 = 15 is not prime.<\/p>\n

So finding primes is a matter of trying higher values of x in the expression 2x<\/i><\/sup> − 1 and testing if they can be factored.<\/p>\n

What is a prime?<\/h2>\n

A prime has exactly 2 factors, itself and 1. So the first prime is the number 2 (factors 2 and 1).<\/p>\n

Who Cares?<\/h2>\n

Every time you use a password to access a bank account or web site, there is a good chance that a large prime number was used in the encryption process. <\/p>\n

Want to make some money with primes?<\/h2>\n

The Electronic Frontier Association paid $100,00 for the above prime, and are offering $150,000 for a prime with 100 million digits and $250,000 for one with 1 billion digits.<\/p>\n

This is where distributed computing power becomes important — according to ZDNet<\/a>, a single computer would need 500 years to test a billion-digit prime number.<\/p>\n

Be the first to comment<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

A 13 million-digit prime was found in 2008. Primes are important in computer security, and if you find one, you can make good money.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[127],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/2714"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=2714"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/2714\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=2714"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=2714"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=2714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}