# Prime Numbers in Linear Patterns

Prime numbers in table with 180 columns
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, which implies that the prime numbers only appear on certain columns.

# 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

## 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

## Properties

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 $p \equiv 1\pmod {30}$, or there exists a prime number $q$ less than 30 such that $p \equiv q \pmod {30}$.

Proof.

Given any prime number $p$, assume that $p$ is neither congruent to $1 \pmod {30}$ nor $q \pmod {30}$ for every prime $q$ less than $30$. Then $p$ is congruent to $x \pmod {30}$, where $x$ is some integer less than $30$ that is not $1$ and not a prime. Prime factorization of $x$ must contain one of $2, 3,$ and $5$. (If the prime factorization of $x$ did not contain any of $2, 3,$ or $5$, then the smallest possible value of $x$ will be $7 \cdot 7 =49$, which is greater than $30$). Thus, $x=2^a3^b5^c$, where $a,b,c \ge 0$, and at least one of $a,b,c$ is greater than $0$.

Since $p$ is congruent to $x$, we can write $p$ as $p=30n+2^a3^b5^c$, where $n$ is an integer greater than or equal to 1. Then, $p=30n+2^a3^b5^c=(2 \cdot 3 \cdot 5)n+2^a3^b5^c$. $p$ is then equal to one of $2(3 \cdot 5 \cdot n+2^{a-1}3^b5^c)$ or $3(2 \cdot 5 \cdot n+2^a3^{b-1}5^c)$ or $5(2 \cdot 3 \cdot n+2^a3^b5^{c-1})$, which contradicts $p$ being a prime number. Thus, $p \equiv 1 \pmod {30}$ or $p \equiv q \pmod {30}$.$\Box$

However, the statement does not generalize to other integer modulo groups. For instance, consider a table with 60 columns. The number 49 appears on the first row, and 49 is not a prime number. However, the column containing 49 will contain other prime numbers, such as 109.

Moreover, not all integers that are congruent to $1 \pmod {30}$ or $q \pmod {30}$, where $q$ is a prime number less than $30$, are prime numbers. For instance, $49$, which is congruent to $19 \pmod {30}$, is not a prime number, but $49$ still appears on the same column as$1$. Let's call the columns that have a 1 or a prime number on its top row as prime-concentrated columns. One can observe that all composite numbers that appear on these prime-concentrated columns, say $49,77,91,119,121,133,143,161,169,...,$ have prime factors that are greater than or equal to $7$. In other words, these composite numbers do not have $2, 3,$or $5$ as a prime factor.

Theorem 2. Composite numbers that appear on prime-concentrated columns do not have $2,3,$ or $5$ as a prime factor.

Proof.

Let $x$ be a composite number that appears on a prime-concentrated column, and assume that $x$ has at least one of $2, 3,$ or $5$ as a prime factor. Since $x$ appears on the prime-concentrated column, $x$ can be written as $x=30n+k$, where $n$ is a positive integer and $k$ is a prime number smaller than $30.$ If $x$ had $2$ as a prime factor, $k$ must also have $2$ as a factor because $30$ has $2$ as a factor. This contradicts the fact that $k$ is a prime number. The same argument works for the case when $x$ has $3$ or $5$ as prime factors.$\Box$

Another pattern to notice is that the prime-concentrated columns seem symmetric about the column that contains $15$, which leads to the following observation.

Theorem 3. If $p$ is a prime number less than$30$ and if $p$ is not equal to $2,3,$ or $5,$ then $30-p$ is a prime number.

Proof.

Let $p$ be a prime number less than $30$ that is not equal to $2, 3,$or $5$. Let $q=30-p$. If $q$ were not a prime, then $q$ must have $2, 3,$ or $5$ as a prime factor. Since $p=30-q$, $p$ will also be divisible by $2, 3,$ or $5$, contradicting our condition that $p$ is a prime number. $\Box$

Such observation triggers one's interest to see whether the above statement is true for any multiples of $30$, or for any number that is a product of the first few primes. However, this turns out not to be the case.

# Teaching Materials

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Q: There seems to be infinitely many primes in each congruence class (i.e, there are infinitely many primes in each prime-concentrated columns)

This is what Dirichlet's theorem on arithmetic progressions is saying

Q: would it be possible to generalize the above statements to any subgroup of the integers modded by the product of first n primes?

i.e, can we generalize above statements to the case where we create a table with more number of columns?

Have questions about the image or the explanations on this page?