# Chen Prime Number Theorem

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Chen's prime number theorem has been invaluable in the study of prime numbers and number theory. This lesson will introduce the theorem, solidify our understanding of it, and briefly discuss the areas in which it has been most significant.

## Chen Prime Number Theorem

If you think about prime numbers, or numbers that are only divisible by one and themselves, you will probably find these types of numbers to be one of the most mysterious, yet fascinating, types of numbers in the history of mathematics. There are a lot of open conjectures, or conjectures that have yet to be proven, that are long-standing due to the fact that it is hard to find patterns that prime numbers follow.

That being said, facts about primes that have been proven do exist, and they aid in the study of prime numbers. One such theorem is the Chen prime number theorem. This theorem was discovered and proven by Chen Jingrun.

In this theorem, a semiprime number is a number that is a product of two primes. In other words, Chen's theorem states that as the even numbers grow larger and larger, eventually they will reach a point when all of them can be written as p + q or p + qr, where p, q, and r are prime numbers. Though we may not know where that point is, we know that such a point exists by Chen's prime number theorem.

To illustrate an even number that satisfies this theorem, consider the number 22. We can write the number 22 as a sum of two primes in two ways:

• 22 = 3 + 19 = 5 + 17

We can also write the number 22 as a sum of a prime and a semiprime in three ways:

• 22 = 7 + 3×5 = 13 + 3×3 = 11 + 2×5

Well, this is neat! We've just explored a pattern that prime numbers follow, so we know that some do exist! So, where has this theorem helped in the study of prime numbers?

## The Study of Prime Numbers

Chen's prime number theorem is significant in that it has helped to bring mathematicians closer to proving some of those unproven conjectures about prime numbers that we've mentioned. One big one is Goldbach's conjecture, which is credited to Christian Goldbach and was brought about in a correspondence between him and another mathematician, Leonhard Euler in 1742.

Notice that this is named Goldbach's conjecture and not Goldbach's theorem. This is because it has yet to be proven. It is one of the most famous unsolved problems in mathematics. Although it remains to be proven definitively, Chen's prime number theorem has contributed greatly to the study of this conjecture and has advanced mathematicians toward a solution.

Chen's prime number theorem has also been quite useful in the study of number theory in areas such as sieve theory, which in simplistic terms, is a way of counting certain sets of integers. One such set is the set of prime numbers. Chen's prime number theorem has contributed in the quest to estimate the number of prime numbers up to various numbers or limits.

## Solidifying our Understanding

Although the mathematics that goes into Chen's prime number theorem and how it is used in the study of number theory is very advanced and well outside the scope of this lesson, we can look at some different even numbers and pick them apart to show that they satisfy the theorem.

We'll start with an easy one. Consider the even number 6. Can you think of two numbers, p and q, that add up to 6, such that either they are both prime or one is prime and the other is a product of two primes? If you're thinking of 3 + 3 or 2 + 4, then you're getting the hang of this!

• 6 = 3 + 3
• 6 = 2 + 4

We can write 6 as the sum 3 + 3, where 3 is a prime number, or we can also write it as 2 + 4, where 2 is a prime number, and 4 is a semiprime since it is the product of 2 and 2 (two prime numbers).

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