Prime Numbers in Linear Patterns

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|ImageIntro=Prime numbers marked in a table with 180 columns
|ImageIntro=Prime numbers marked in a table with 180 columns
|ImageDescElem=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.
|ImageDescElem=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.
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|ImageDesc=Instead of studying a table with 180 columns, we will study a table with 30 columns.
+
|ImageDesc=Instead of studying a table with 180 columns, we will study a table with 30 columns, as shown in [[#1|Image 1]].
{{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"
+
""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 [[#2|Image 2]].
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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either p � 1 (mod 30), or there exists a prime number q less than 30
either p � 1 (mod 30), or there exists a prime number q less than 30
such that p � q (mod 30).
such that p � q (mod 30).
-
 
|AuthorName=Iris Yoon
|AuthorName=Iris Yoon
|Field=Algebra
|Field=Algebra
|InProgress=No
|InProgress=No
}}
}}

Revision as of 00:05, 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, as shown in [[#1 [...]

Instead of studying a table with 180 columns, we will study a table with 30 columns, as shown in Image 1.

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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