# Prove that primes of the form 4n+1 are infinite

• tamalkuila
In summary, there are an infinite number of primes of the form 4n+1, meaning that there are an infinite number of primes that can be expressed as one more than four times an integer. This can be proven by contradiction, assuming there is a finite number of such primes, and showing that this leads to a contradiction. The email given is tamalkuila@gmail.com.
tamalkuila
prove that primes of the form 4n+1 are infinite . send the proof at tamalkuila@gmail.com

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Hmmmm... Try doing a proof by contradiction.

Why would 4n+1 be infinite? n is an unknown variable. The only way the statement $$4n+1$$ is true is if $$n = infity$$.

PS >> Sorry I did not remember the \infity thing, it didnt work

Last edited:
Good point, eNathan, but I think the original poster mean "the number of primes of the form 4n+1 is infinite".

Im a little confused with the logic of this question. What do prime numbers have to do with 4n+1?

A prime of the the form 4n+1 is a prime that is equal to one more than four times an integer. In other words the prime when divided by four has a remainder of 1.

5, 13, 17, 29 are examples of primes of the form 4n+1.

5=4(1)+1
13=4(3)+1
17=4(4)+1
29=4(7)+1

## What is the significance of primes of the form 4n+1?

Primes of the form 4n+1 are important because they have unique properties that make them useful in various mathematical proofs and applications.

## How can you prove that primes of the form 4n+1 are infinite?

The proof for this statement is based on the assumption that there is a finite number of primes of the form 4n+1. By using this assumption, a contradiction can be reached, proving that there must be an infinite number of primes of this form.

## Can you provide an example of a proof for this statement?

One example of a proof for this statement is the classic proof by contradiction, also known as Euclid's proof. It involves assuming that there is a finite number of primes of the form 4n+1 and then constructing a number that is larger than any of these primes, ultimately showing that there must be an additional prime of this form. This proof was first published by Euclid in his book "Elements" in 300 BC.

## What are some other methods for proving this statement?

There are several other methods for proving that primes of the form 4n+1 are infinite, including the use of modular arithmetic and number theory. Some of these methods involve more advanced mathematical concepts and may require a deeper understanding of mathematics to fully comprehend.

## Why is it important to prove that primes of the form 4n+1 are infinite?

Proving that primes of the form 4n+1 are infinite has implications in many areas of mathematics, including number theory, cryptography, and computer science. It also helps in understanding the distribution and properties of prime numbers, which have been studied for centuries and continue to be a topic of interest for mathematicians and scientists.

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