Are there an infinite amount of prime numbers

Click the photo for a link to the amazon page, or this link for the ebook. Email Address. Skip to content. Home About Faq. Q: How plausible is it that the laws of physics may actually function differently in other parts of the universe? Q: Are there an infinite number of prime numbers? Posted on October 21, by The Physicist. Email Print Facebook Reddit Twitter.

This entry was posted in -- By the Physicist , Math. Bookmark the permalink. David says:. January 25, at pm. October 2, at am. Dylan says:. For example, if we begin with the set:. The new prime found would be , , or , all of which are smaller than the last prime in the original set. Euclid's Proof of the Infinitude of Primes c.

There are infinitely many primes. So this prime p is still another prime, and p 1 , p 2 , There are more primes than found in any finite list of primes. Call the primes in our finite list p 1 , p 2 , Now P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. It will never finish the task! So the brute force approach is out of question. This is where we use something called proof by contradiction.

Proof by contradiction is a popular technique used to prove results. What we will do is we are going to assume that the opposite is true, and then show that it leads to a fundamental contradiction. Then, we can assert that the opposite must be true. We are going to assume that we have a finite number of primes and see where it takes us. This would imply that there is a largest prime number, after which all numbers are composite.

So now, we have a list out all the prime numbers that exist: 2, 3, 5, 7, …, L. As we can see, this is a huge number, but it is still finite.

So this means that M is a composite number. If M is a prime number, then L would cease to be the largest prime number. This would be a contradiction! Okay so we established that M is not a prime number. This would mean that M must be divisible by a prime number. Every composite number can factorized into its prime factors.

For example, the number 18 can be written as 2 x 3 x 3. As we can see, both 2 and 3 are prime numbers. This holds true for every single composite number. Since we know M is not prime, it must be divisible by a prime number.

This is because the first term in M i. So after this, the second term i.