{"id":4260,"date":"2010-03-08T10:48:23","date_gmt":"2010-03-08T02:48:23","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=4260"},"modified":"2015-01-12T20:40:44","modified_gmt":"2015-01-12T12:40:44","slug":"lines-of-primes","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/lines-of-primes-4260","title":{"rendered":"Lines of primes"},"content":{"rendered":"

Prime numbers have fascinated mathematicians for centuries. A prime number has exactly 2 factors - one and itself. The only even prime is 2, the rest are all odd. <\/p>\n

The primes less than 100 are as follows: <\/p>\n

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97<\/p>\n

There doesn't appear to be a pattern in the distribution of primes.<\/p>\n

How about the "gap" (spacing) between the primes? Is there a pattern in that?<\/p>\n

1\t2\t2\t4\t2\t4\t2\t4\t6\t2\t6\t4\t2\t4\t6\t6\t2\t6\t4\t2\t6\t4\t6\t8<\/p>\n

There doesn't appear to be a pattern in the gaps, either.<\/p>\n

Spiraling <\/h2>\n

Stanislaw Ulam was a Polish-American mathematician who was involved in the Manhattan Project during World War II.<\/p>\n

One day he was bored in a meeting and began to write numbers in a spiral. He started like this, moving in a clockwise direction. <\/p>\n\n\n\n
1 →<\/td>\n2 ↓<\/td>\n<\/tr>\n
4 ← <\/td>\n3<\/td>\n<\/tr>\n<\/table>\n

The next round continued the "spiraling" pattern, as follows. <\/p>\n\n\n\n\n
7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n<\/tr>\n
6<\/td>\n1<\/td>\n2<\/td>\n11<\/td>\n<\/tr>\n
5<\/td>\n4<\/td>\n3<\/td>\n12<\/td>\n<\/tr>\n<\/table>\n

He kept going (it must have been a long meeting), then highlighted the prime numbers and found something interesting.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
73<\/td>\n74<\/td>\n75<\/td>\n76<\/td>\n77<\/td>\n78<\/td>\n79<\/td>\n80<\/td>\n81<\/td>\n82<\/td>\n<\/tr>\n
72<\/td>\n43<\/td>\n44<\/td>\n45<\/td>\n46<\/td>\n47<\/td>\n48<\/td>\n49<\/td>\n50<\/td>\n83<\/td>\n<\/tr>\n
71<\/td>\n42<\/td>\n21<\/td>\n22<\/td>\n23<\/td>\n24<\/td>\n25<\/td>\n26<\/td>\n51<\/td>\n84<\/td>\n<\/tr>\n
70<\/td>\n41<\/td>\n20<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n27<\/td>\n52<\/td>\n85<\/td>\n<\/tr>\n
69<\/td>\n40<\/td>\n19<\/td>\n6<\/td>\n1<\/td>\n2<\/td>\n11<\/td>\n28<\/td>\n53<\/td>\n86<\/td>\n<\/tr>\n
68<\/td>\n39<\/td>\n18<\/td>\n5<\/td>\n4<\/td>\n3<\/td>\n12<\/td>\n29<\/td>\n54<\/td>\n87<\/td>\n<\/tr>\n
67<\/td>\n38<\/td>\n17<\/td>\n16<\/td>\n15<\/td>\n14<\/td>\n13<\/td>\n30<\/td>\n55<\/td>\n88<\/td>\n<\/tr>\n
66<\/td>\n37<\/td>\n36<\/td>\n35<\/td>\n34<\/td>\n33<\/td>\n32<\/td>\n31<\/td>\n56<\/td>\n89<\/td>\n<\/tr>\n
65<\/td>\n64<\/td>\n63<\/td>\n62<\/td>\n61<\/td>\n60<\/td>\n59<\/td>\n58<\/td>\n57<\/td>\n90<\/td>\n<\/tr>\n
100<\/td>\n99<\/td>\n98<\/td>\n97<\/td>\n96<\/td>\n95<\/td>\n94<\/td>\n93<\/td>\n92<\/td>\n91<\/td>\n<\/tr>\n<\/table>\n

Many of the primes appear to line up when arranged in such a sprial.<\/p>\n

Let's go much bigger and see what happens. We observe there are many places where the primes form line segments, mostly at 45°, but sometimes horizontal and vertical.<\/p>\n

\"prime<\/p>\n

What I found interesting in the large picture is where primes are not<\/strong> - there are distinct blocks and patterns of white space where no primes occur. <\/p>\n

This spiral appeared on the cover of Scientific American in March 1964 and continues to generate research interest to this day.<\/p>\n

Why do we care about primes?<\/h2>\n

Apart from many other things, prime numbers are vital in the development of encryption algorithms, used in generating secure Internet transactions. <\/p>\n

See the 6 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

\"primes\"<\/a>Prime numbers seem to be randomly distributed — but perhaps there are patterns after all.<\/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":[134],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4260"}],"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=4260"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/4260\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=4260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=4260"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=4260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}