Chen prime facts for kids

In mathematics, a Chen prime is a special kind of prime number. A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself (like 2, 3, 5, 7). A prime number p is called a Chen prime if the number p + 2 is either another prime number or a semiprime. A semiprime is a number that is the product of exactly two prime numbers (for example, 6 is a semiprime because 2 × 3 = 6).

These special primes are named after a brilliant Chinese mathematician named Chen Jingrun. In 1966, he proved a very important idea: that there are infinitely many Chen primes. This means that no matter how many you find, there will always be more!

What Are Chen Primes?

A Chen prime is a prime number p where the number p + 2 fits one of two rules:

• It is also a prime number.
• It is a semiprime (a number made by multiplying just two prime numbers together).

For example, 3 is a Chen prime because 3 + 2 = 5, and 5 is a prime number. Another example is 11. 11 + 2 = 13, and 13 is also a prime number. What about 19? 19 + 2 = 21. 21 is not a prime number (it's 3 × 7). But since 21 is the product of two prime numbers (3 and 7), it is a semiprime. So, 19 is also a Chen prime!

Who Was Chen Jingrun?

Chen Jingrun, the mathematician who proved there are infinitely many Chen primes.

Chen Jingrun was a famous Chinese mathematician. He was born in 1933 and passed away in 1996. His work on prime numbers, especially the one that led to Chen primes, was a huge step forward in understanding these mysterious numbers. His proof in 1966 showed that there are endless Chen primes, which was a big achievement in number theory.

Examples of Chen Primes

Here are the first few Chen primes:

• 2 (because 2 + 2 = 4, and 4 = 2 × 2, which is a semiprime)
• 3 (because 3 + 2 = 5, which is a prime)
• 5 (because 5 + 2 = 7, which is a prime)
• 7 (because 7 + 2 = 9, and 9 = 3 × 3, which is a semiprime)
• 11 (because 11 + 2 = 13, which is a prime)
• 13 (because 13 + 2 = 15, and 15 = 3 × 5, which is a semiprime)
• 17 (because 17 + 2 = 19, which is a prime)
• 19 (because 19 + 2 = 21, and 21 = 3 × 7, which is a semiprime)
• 23 (because 23 + 2 = 25, and 25 = 5 × 5, which is a semiprime)
• 29 (because 29 + 2 = 31, which is a prime)
• 31 (because 31 + 2 = 33, and 33 = 3 × 11, which is a semiprime)
• 37 (because 37 + 2 = 39, and 39 = 3 × 13, which is a semiprime)
• 41 (because 41 + 2 = 43, which is a prime)
• 47 (because 47 + 2 = 49, and 49 = 7 × 7, which is a semiprime)
• 53 (because 53 + 2 = 55, and 55 = 5 × 11, which is a semiprime)
• 59 (because 59 + 2 = 61, which is a prime)
• 67 (because 67 + 2 = 69, and 69 = 3 × 23, which is a semiprime)
• 71 (because 71 + 2 = 73, which is a prime)
• 83 (because 83 + 2 = 85, and 85 = 5 × 17, which is a semiprime)
• 89 (because 89 + 2 = 91, and 91 = 7 × 13, which is a semiprime)
• 101 (because 101 + 2 = 103, which is a prime)

... and many more!

Chen Primes and Twin Primes

A special type of prime pair is called a twin prime. Twin primes are pairs of prime numbers that are only two apart, like (3, 5) or (5, 7). The Twin prime conjecture is a famous idea in math that suggests there are infinitely many such pairs. If this conjecture is true, it would mean that the smaller number in every twin prime pair is a Chen prime. This is because if p and p + 2 are both prime, then p is a Chen prime by definition.

Here are some Chen primes that are not the smaller number of a twin prime pair (meaning p + 2 is a semiprime, not a prime):

• 2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... (sequence A063637 in OEIS)

Non-Chen Primes

Not all prime numbers are Chen primes. Here are some examples of prime numbers that are not Chen primes:

• 43 (because 43 + 2 = 45, and 45 = 3 × 3 × 5, which is not a prime or a semiprime)
• 61 (because 61 + 2 = 63, and 63 = 3 × 3 × 7, which is not a prime or a semiprime)
• 73 (because 73 + 2 = 75, and 75 = 3 × 5 × 5, which is not a prime or a semiprime)
• 79 (because 79 + 2 = 81, and 81 = 3 × 3 × 3 × 3, which is not a prime or a semiprime)
• 97 (because 97 + 2 = 99, and 99 = 3 × 3 × 11, which is not a prime or a semiprime)

... and so on.

Interesting Facts About Chen Primes

The Magic Square of Chen Primes

A magic square is a grid of numbers where the numbers in each row, column, and main diagonal all add up to the same total. Rudolf Ondrejka, a mathematician, found a 3 × 3 magic square made entirely of Chen primes!

In this square, each row, column, and main diagonal adds up to 177. For example, 17 + 89 + 71 = 177.

Discovering the Largest Chen Prime

Mathematicians and computer scientists are always searching for larger and larger prime numbers. As of 2025, the largest known Chen prime is a massive number: 2996863034895 × 2^1290000 − 1. This number has an incredible 388342 digits! Finding such huge primes often requires powerful computers and advanced algorithms.

More Discoveries About Chen Primes

Chen's General Rule

Chen Jingrun didn't just prove the rule for p + 2. He also showed a more general idea. For any even whole number h (like 2, 4, 6, etc.), there are infinitely many prime numbers p such that p + h is either a prime number or a semiprime. This means the idea of Chen primes can be extended to many other number relationships!

Patterns in Chen Primes

Mathematicians love to find patterns. Ben Green and Terence Tao, two famous mathematicians, discovered in 2004 that Chen primes contain infinitely many arithmetic progressions of length 3. An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. For example, 3, 5, 7 is an arithmetic progression with a difference of 2. Later, another mathematician named Binbin Zhou expanded on this, showing that Chen primes can form arithmetic progressions of any length! This means you can find sequences of Chen primes like a, a + d, a + 2d, a + 3d, and so on, for as long as you want.