Twin prime

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). This means a twin prime is a prime number that has a prime gap of two. Sometimes the term twin prime refers to the pair itself, also called a prime twin or prime pair.

As numbers get larger, twin primes become rarer because the gaps between prime numbers usually grow wider. One of the big questions in mathematics is whether there are infinitely many twin primes, known as the twin prime conjecture. This question is still unsolved.

Important progress was made in 2013 by Yitang Zhang, and later by James Maynard, Terence Tao, and others. Their work has moved us closer to proving that there are infinitely many twin primes, but we do not yet have a final answer. This remains one of the interesting unsolved problems in mathematics, as shown in the list of more unsolved problems in mathematics.

Properties

Twin primes are pairs of prime numbers that are very close to each other—specifically, they differ by exactly 2. For example, 17 and 19 are twin primes because both are prime and only 2 apart. Usually, the pair (2, 3) is not counted as twin primes because they differ by just 1, and 2 is the only even prime number.

Some early examples of twin primes include (3, 5), (5, 7), (11, 13), (17, 19), and (29, 31). An interesting fact is that the number 5 is special—it belongs to two twin prime pairs. Most twin prime pairs follow a pattern: they can be written in the form (6n – 1, 6n + 1) for some whole number n. This means the number between the two primes is always a multiple of 6. Because of this pattern, the sum of any twin prime pair (except for 3 and 5) is always divisible by 12.

Brun's theorem

Main article: Brun's theorem

In 1915, a mathematician named Viggo Brun proved something important about twin primes. He showed that if you add up the reciprocals (or flips) of all the twin primes, the total amount stays smaller and smaller as you include more twin primes. This discovery was one of the first uses of a special math tool called the Brun sieve and helped start the study of modern sieve theory.

Twin prime conjecture

The twin prime conjecture is a big question in math that asks if there are infinitely many twin primes. Twin primes are pairs of prime numbers that are two apart, like 17 and 19. This question has been open for many years.

In 2013, a mathematician named Yitang Zhang made a big step forward by showing that there are infinitely many pairs of primes that differ by some number less than 70 million. Later, this number was made even smaller, down to 246. This work helps mathematicians understand how common twin primes might be.

Other theorems weaker than the twin prime conjecture

In 1940, mathematician Paul Erdős showed that there is a special number, called C₂, connected to prime numbers. This number helps us understand how often twin primes might appear, although it doesn’t prove that there are infinitely many of them.

Another idea, called Polignac’s conjecture from 1849, suggests that for every even number, there are infinitely many pairs of primes that differ by that number. The twin prime conjecture is a special case of this, where the number is 2. While this hasn’t been proven yet, a discovery by Zhang shows that at least one such even number must exist, even though we don’t know which one it is.

Large twin primes

Since 2007, two special computer projects called Twin Prime Search and PrimeGrid have found some of the biggest twin primes known. As of January 2025, the largest twin prime pair discovered is 2996863034895 × 2^1290000 ± 1, which has 388,342 digits. This amazing discovery was made in September 2016.

Scientists have counted that there are 808,675,888,577,436 twin prime pairs below 10¹⁸. They also study how often these pairs appear and have found patterns in their numbers.

Other elementary properties

Did you know that every third odd number can’t be prime unless it’s the number 3? This means that 5 is the only prime number that can be part of two twin prime pairs. Also, if you have three prime numbers very close together—like m − 4, m, and m + 6—they are called a prime triplet.

There are special patterns for twin primes of the form (6ⁿ − 1, 6ⁿ + 1). For these pairs to work, the number n has to end in certain digits like 0, 2, 3, 5, 7, or 8. If n ends in 1, 4, 6, or 9, one of the numbers in the pair won’t be prime. This helps mathematicians find and study twin primes more easily.

Main article: Chen prime
Main article: prime triplet

Isolated prime

An isolated prime is a prime number that has no other prime number just two away from it — meaning neither one less nor one more than it is also prime. For example, 23 is an isolated prime because 21 and 25 are not prime numbers.

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