Prime Numbers in Linear Patterns

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Prime numbers in table with 180 columns
Screen shot 2012-10-01 at 2.49.15 PM.png
Field: Algebra
Image Created By: Iris Yoon

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.

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

Image 1

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 Image 1. He then circled the prime numbers, and the prime numbers showed patterns of diagonal lines as shown by the grid in Image 2. The grid in 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.

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.

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