Cousin primes are prime pairs separated by exactly 4. The pair (7, 11) is a simple example because both numbers are prime and 11 − 7 = 4.
Definition
A cousin prime pair is a pair of prime numbers in the form (p, p + 4). Both numbers must be prime, and the difference between them must be exactly 4.
What Cousin Primes Mean
A prime number is a whole number greater than 1 with exactly two positive divisors: 1 and itself. Cousin primes use that same definition twice. The lower number must be prime, and the number four units higher must also be prime.
For example, 13 and 17 form a cousin prime pair because 13 is prime, 17 is prime, and their difference is 4. The number 15 does not affect the pair because the definition is about the two selected primes, not every number between them.
Cousin Prime Form
(p, p + 4)
Here, p is prime and p + 4 is also prime.
Examples of Cousin Prime Pairs
The first few cousin prime pairs show the pattern clearly. Each row below contains two prime numbers with a difference of 4.
| Cousin Prime Pair | Difference | Why It Qualifies |
|---|---|---|
| (3, 7) | 4 | Both 3 and 7 are prime. |
| (7, 11) | 4 | Both 7 and 11 are prime. |
| (13, 17) | 4 | Both 13 and 17 are prime. |
| (19, 23) | 4 | Both 19 and 23 are prime. |
| (37, 41) | 4 | Both 37 and 41 are prime. |
| (43, 47) | 4 | Both 43 and 47 are prime. |
| (67, 71) | 4 | Both 67 and 71 are prime. |
| (79, 83) | 4 | Both 79 and 83 are prime. |
Cousin Primes and Prime Gaps
A prime gap is the difference between consecutive prime numbers. Cousin primes are related to prime gaps, but the two ideas are not always identical.
Common Mistake
Every cousin prime pair differs by 4, but not every cousin prime pair is a prime gap of 4. The pair (3, 7) differs by 4, but 5 is also prime and lies between them. So (3, 7) is a cousin prime pair, not a gap of 4 between consecutive primes.
The pair (7, 11) is different. There is no prime number between 7 and 11, so it is both a cousin prime pair and a prime gap of 4.
Why the Difference Is Always Even After the Smallest Cases
After 2, every prime number is odd. The difference between two odd numbers is even, so prime pairs beyond the smallest prime naturally have even separation. Cousin primes use the even difference 4.
There is also a useful modular pattern. Every prime greater than 3 is either one less than a multiple of 6 or one more than a multiple of 6. For cousin primes greater than 3, the pair fits this form:
Common Candidate Pattern
(6n + 1, 6n + 5)
This pattern can find candidates, but it does not prove that both numbers are prime.
For example, when n = 1, the pair is (7, 11), and both are prime. When n = 5, the pair is (31, 35), but 35 is composite. A candidate pattern narrows the search; it does not replace a primality test.
Math Note
To verify a possible cousin prime pair, each number must be tested for primality. You can check whether a number is prime before comparing it with the number four units away.
Cousin Primes Compared With Nearby Prime Pair Types
Prime pairs are often named by the distance between the two primes. The distance matters because it changes the arithmetic pattern and the way the pair relates to prime gaps.
| Pair Type | Difference | Example | Basic Idea |
|---|---|---|---|
| Twin primes | 2 | (11, 13) | Two primes separated by 2. |
| Cousin primes | 4 | (13, 17) | Two primes separated by 4. |
| Prime pair separated by six | 6 | (23, 29) | Two primes separated by 6. |
Are There Infinitely Many Cousin Primes?
Mathematicians have not proved whether there are infinitely many cousin prime pairs. Many examples are known, and they continue to appear as numbers grow, but a proof for infinitely many pairs with difference 4 is not known.
This is part of a broader question about fixed prime gaps. The idea that every even gap occurs infinitely often between primes is a famous open direction in number theory. A proof for gap 4 would settle the cousin prime case, but that proof is not currently available.
Known
There are many verified cousin prime pairs.
Not Proven
It is not known whether infinitely many cousin prime pairs exist.
Candidate Rule
The form (6n + 1, 6n + 5) helps filter candidates greater than 3.
Related Concepts
FAQ About Cousin Primes
What Are Cousin Primes?
Cousin primes are two prime numbers with a difference of exactly 4. The pair (19, 23) is an example because both numbers are prime and 23 − 19 = 4.
Is (3, 7) a Cousin Prime Pair?
Yes. Both 3 and 7 are prime, and their difference is 4. However, they are not consecutive primes because 5 lies between them.
Are Cousin Primes the Same as Prime Gaps of 4?
Not always. A cousin prime pair only requires two primes to differ by 4. A prime gap of 4 requires the two primes to be consecutive, with no other prime between them.
What Is the First Cousin Prime Pair?
The first cousin prime pair is (3, 7). The next examples include (7, 11), (13, 17), and (19, 23).
Do All Numbers in the Form (6n + 1, 6n + 5) Make Cousin Primes?
No. That form gives possible candidates greater than 3, but one or both numbers can still be composite. For example, (31, 35) fits the form, but 35 is not prime.
Has It Been Proved That Infinitely Many Cousin Primes Exist?
No. It remains unproved whether infinitely many cousin prime pairs exist. The known examples support the pattern as a topic of study, but examples alone do not prove an infinite result.