# Twin Primes and Brun's Constant

• Diffy
In summary, if Brun's constant is irrational, it would imply that there are an infinite number of primes, as it is the sum of a finite number of fractions. The fact that Brun's constant would be irrational would also confirm the infinite nature of twin primes. However, showing that Brun's constant is irrational may be more difficult than proving the infinitude of twin primes, unless an equivalent expression for Brun's constant is found.
Diffy
If one could show that Brun's constant is irrational, would that imply that there are an infinite number of primes?

I think it would since Brun's constant is the sum of a bunch of fractions, and the sum of a finite number of fractions must be rational. Thus is the sum is irrational there must not be a finite number of fractions...

Is my thinking correct?

Yes, if Brun's constant were to be shown irrational, then there are an infinite number of twin primes, and hence primes. Yes your thinking is correct.

Unless you find some equivalent expression that doesn't use the infinite sum of reciprocals of twin primes which computes Brun's constant B, it will be more difficult to show that B is irrational than it is to show that twin primes are infinite, I think.

## 1. What are twin primes?

Twin primes are a pair of prime numbers that are only two numbers apart from each other, such as 41 and 43 or 71 and 73. They are the closest possible prime numbers and are always odd.

## 2. What is Brun's constant?

Brun's constant is a mathematical constant named after French mathematician Viggo Brun. It is denoted by the letter B and is approximately equal to 1.9021605. It is used in the study of twin primes and the twin prime conjecture.

## 3. What is the twin prime conjecture?

The twin prime conjecture states that there are infinitely many twin primes. In other words, there are an infinite number of pairs of prime numbers that are only two numbers apart from each other.

## 4. Are twin primes important in mathematics?

Yes, twin primes are important in mathematics because they provide insight into the distribution of prime numbers. They also play a role in cryptography and number theory.

## 5. Is the twin prime conjecture proven or still a conjecture?

The twin prime conjecture is still a conjecture and has not been proven. However, there has been significant progress made towards proving it, with the latest result being that there are infinitely many primes that are at most 246 apart from each other.

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