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]]}} | ||
- | + | ==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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{{Anchor|Reference=2|Link=[[Image:Irisprime.jpg|Image 2|thumb|700px|none]]}} | {{Anchor|Reference=2|Link=[[Image:Irisprime.jpg|Image 2|thumb|700px|none]]}} | ||
- | All prime numbers appear on columns that have a 1 or a prime | + | ==Theorem 1. 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 :<math>p \equiv 1 \pmod 30</math>, or there exists a prime number q less than 30 such that :<math> p \equiv q \pmod 30</math>. == |
- | number on its top row. In other words, for every prime number p, | + | |
- | either p | + | |
- | such that p | + | |
|AuthorName=Iris Yoon | |AuthorName=Iris Yoon | ||
|Field=Algebra | |Field=Algebra | ||
|InProgress=No | |InProgress=No | ||
}} | }} |
Revision as of 00:11, 5 December 2012
Prime numbers in table with 180 columns |
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Prime numbers in table with 180 columns
- Prime numbers marked in a table with 180 columns
Contents |
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.
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.
Theorem 1. 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 :, or there exists a prime number q less than 30 such that :.
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