Twin Primes and Brun's Constant

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SUMMARY

This discussion centers on the implications of Brun's constant being irrational in relation to the infinitude of twin primes. Participants agree that if Brun's constant, defined as the sum of the reciprocals of twin primes, is proven to be irrational, it would indicate the existence of an infinite number of twin primes. The conversation highlights the challenge of demonstrating the irrationality of Brun's constant compared to proving the infinitude of twin primes directly.

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  • Understanding of twin primes and their properties
  • Familiarity with Brun's constant and its mathematical definition
  • Knowledge of rational vs. irrational numbers
  • Basic concepts of infinite series and sums in mathematics
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Mathematicians, number theorists, and students interested in prime number theory and the properties of mathematical constants.

Diffy
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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?
 
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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.
 

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