# Number theory: confused about the phrase an integer of the form

• doubleaxel195
In summary: In Case 1, 3k+1 can't be written as 6m+1 because 3k+1 is odd. In Case 2, 3k+1 can be written as 6m+1 because 3k+1 is even.

#### doubleaxel195

Number theory: confused about the phrase "an integer of the form"

## Homework Statement

Prove that any prime of the form 3k+1 is of the form 6k+1

## The Attempt at a Solution

I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But that gets me no where really, and doesn't utilize the fact that it must be prime and now I made my prime number not prime.

But first and foremost, I am confused about the phrase/notation "an integer of the form." For instance, in this problem, does it mean that k must have the same value or are those 2 different k's where k is just any integer but not necessarily the same integer. Well, obviously for k=1, 3(1)+1=4 whereas, 6(1)+1=7 so they are not equal. So why is the same variable k used? Or is k used to refer to the set of values that 3k+1 and 6k+1 can take on? Like the set of values for 6k+1 is contained within 3k+1, like a subset. If this is true, then do I just use p=3k+1 and just show that an integer exists, not necessarily k, such that p=6q+1?

doubleaxel195 said:

## Homework Statement

Prove that any prime of the form 3k+1 is of the form 6k+1

## The Attempt at a Solution

I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But that gets me no where really, and doesn't utilize the fact that it must be prime and now I made my prime number not prime.
I don't believe they intended for the k on one side to be the same as the k on the other side. What I think they meant was that any prime of the form 3k + 1 is also of the form 6m + 1, for some integers k and m.
doubleaxel195 said:
But first and foremost, I am confused about the phrase/notation "an integer of the form." For instance, in this problem, does it mean that k must have the same value or are those 2 different k's where k is just any integer but not necessarily the same integer. Well, obviously for k=1, 3(1)+1=4 whereas, 6(1)+1=7 so they are not equal.
Bad example, since 4 isn't a prime.

A better choice is 3(6) + 1 = 19, which is a prime. 19 can also be written as 6(3) + 1.
doubleaxel195 said:
So why is the same variable k used? Or is k used to refer to the set of values that 3k+1 and 6k+1 can take on? Like the set of values for 6k+1 is contained within 3k+1, like a subset. If this is true, then do I just use p=3k+1 and just show that an integer exists, not necessarily k, such that p=6q+1?

If the prime of the form 3k+1 can't be written in the form 6m+1, what does that say about k?

Is k an integer? I can't really see a way out of this. However if you can prove that every 3k + 1 number can be written as 6m + 1 after a certain value of k, then it shouldn't be too difficult.

EDIT: Oops! I think you can show that if k is even, then your 3k + 1 is odd and can be written as 6m + 1, which means that all odd numbers of the form 3k + 1 can be written as 6m +1. Try that.

Split it up into two cases:

Case 1: k is odd. What happens if k is odd? Is this possible?

Case 2: k is even. How can we write k?

## 1. What is an integer of the form?

An integer of the form refers to a number that can be written in a specific way, usually in terms of other numbers or mathematical operations. For example, an integer of the form 2n would be any even number, since it can be written as 2 times some value n.

## 2. Can you give an example of an integer of the form?

Yes, an integer of the form 3n+1 would be any number that, when divided by 3, leaves a remainder of 1. So 4, 7, 10, and 13 are all examples of integers of this form.

## 3. What is the significance of studying integers of the form?

Studying integers of the form can help us understand patterns and relationships between numbers. This can lead to new discoveries and applications in various fields, such as cryptography and computer science.

## 4. How is number theory related to integers of the form?

Number theory is a branch of mathematics that deals with the properties and relationships of integers. Integers of the form are a common topic in number theory, as they can help us understand the behavior of numbers and solve complex problems.

## 5. Are there any famous theorems or results related to integers of the form?

Yes, there are several famous theorems and results related to integers of the form, such as Fermat's Little Theorem and the Chinese Remainder Theorem. These theorems have important applications in various fields and have been studied extensively by mathematicians throughout history.