Twin Prime Conjecture — Definition, Formula & Examples

The Twin Prime Conjecture states that there are infinitely many pairs of prime numbers that differ by exactly 2, such as (11, 13) and (29, 31). Despite strong numerical evidence and centuries of effort, no one has proven or disproven this claim.

The conjecture asserts that the set $\{p \in \mathbb{Z} : p \text{ and } p+2 \text{ are both prime}\}$ is infinite. Equivalently, for every positive integer $N$ , there exists a prime $p > N$ such that $p + 2$ is also prime.

How It Works

Two primes are called twin primes if they differ by 2. The smallest twin prime pairs are

(3,5)

(5,7)

(11,13)

(17,19)

, and

(29,31)

. As numbers grow larger, primes become sparser, yet twin prime pairs keep appearing. Computers have found enormous twin prime pairs, but finding examples—no matter how many—cannot prove the conjecture true for all numbers. In 2013, Yitang Zhang made a breakthrough by proving that infinitely many prime pairs differ by at most 70 million, and subsequent work by the Polymath project reduced this bound to 246, but closing the gap to 2 remains open.

Worked Example

Problem: Verify that (41, 43) is a twin prime pair.

Step 1: Check whether 41 is prime. Test divisibility by primes up to

\sqrt{41} \approx 6.4

, so check 2, 3, and 5.

41 \div 2 = 20.5,\quad 41 \div 3 \approx 13.67,\quad 41 \div 5 = 8.2

Step 2: None divide evenly, so 41 is prime. Now check 43 the same way (primes up to

\sqrt{43} \approx 6.6

43 \div 2 = 21.5,\quad 43 \div 3 \approx 14.33,\quad 43 \div 5 = 8.6

Step 3: 43 is also prime, and the difference is exactly 2.

43 - 41 = 2

Answer: Yes, (41, 43) is a twin prime pair because both numbers are prime and they differ by 2.

Why It Matters

The Twin Prime Conjecture is one of the most famous open problems in mathematics and a staple example in introductory number theory courses. Working with it builds skills in primality testing, proof techniques, and understanding the distribution of primes—topics that underpin modern cryptography and computer science.

Common Mistakes

Mistake: Treating the conjecture as a proven theorem because many twin primes have been found.

Correction: No finite list of examples can prove a statement about infinitely many numbers. The Twin Prime Conjecture remains unproven despite extensive computational evidence.