Prime Numbers in Linear Patterns

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{{Anchor|Reference=1|Link=[[Image:primes30.png|Image 1|thumb|250px|left]]}}
{{Anchor|Reference=1|Link=[[Image:primes30.png|Image 1|thumb|250px|left]]}}
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"Construction"
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The prime spiral was discovered by Stanislaw Ulam (1909-1984) in 1963 while he was doodling on a piece of paper during a science meeting. Starting with 1 in the middle, he wrote positive numbers in a grid as he spiraled out from the center, as shown in [[#1|Image 1]]. He then circled the prime numbers, and the prime numbers showed patterns of diagonal lines as shown by the grid in [[#2|Image 2]]. The grid in [[#2|Image 2]] is a close-up view of the center of the main image such that the green line segments and red boxes in the center of the main image line up with those in the grid.
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First, create a table with 30 columns and sufficiently many rows. Write all positive integers starting from 1 as one moves from left to right, and top to bottom. Then, each row will start with a multiple of 30 added by 1, such as 1, 31, 61, 91, 121, ... . If we mark the prime numbers in this table we get [[#2|Image 2]].
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{{Anchor|Reference=2|Link=[[Image:Irisprime.jpg|Image 2|thumb|700px|none]]}}
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All prime numbers appear on columns that have a 1 or a prime
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number on its top row. In other words, for every prime number p,
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either p � 1 (mod 30), or there exists a prime number q less than 30
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such that p � q (mod 30).
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A larger Ulam spiral with 160,000 integers and 14,683 primes is shown in the main image. Black dots indicate prime numbers. In addition to diagonal line segments formed by the black dots, we can see white vertical and horizontal line segments that cross the center of the spiral and do not contain any black dots, or prime numbers. There are also white diagonal line segments that do not contain any prime numbers. Ulam spiral implies that there is some order in the distribution of prime numbers.
 
|AuthorName=Iris Yoon
|AuthorName=Iris Yoon
|Field=Algebra
|Field=Algebra
|InProgress=No
|InProgress=No
}}
}}

Revision as of 00:04, 5 December 2012


Prime numbers in table with 180 columns

Prime numbers marked in a table with 180 columns


Basic Description

Arranging natural numbers in a particular way and marking the prime numbers can lead to interesting patterns. For example, consider a table with 180 columns and infinitely many rows. Write positive integers in increasing order from left to right, and top to bottom. If we mark all the prime numbers, we get a pattern shown in the figure. We can see that prime numbers show patterns of vertical line segments.

A More Mathematical Explanation

Instead of studying a table with 180 columns, we will study a table with 30 columns.

UNIQ264816974a [...]

Instead of studying a table with 180 columns, we will study a table with 30 columns.

Image 1
Image 1


"Construction"

First, create a table with 30 columns and sufficiently many rows. Write all positive integers starting from 1 as one moves from left to right, and top to bottom. Then, each row will start with a multiple of 30 added by 1, such as 1, 31, 61, 91, 121, ... . If we mark the prime numbers in this table we get Image 2.

Image 2
Image 2


All prime numbers appear on columns that have a 1 or a prime number on its top row. In other words, for every prime number p, either p � 1 (mod 30), or there exists a prime number q less than 30 such that p � q (mod 30).




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