Vipul: Created page with '==Statement== The theorem states the following: For any $x > 0$ , the number of fact about::twin primes less than or equal to $x$ is bounded from above…'

2010-05-29T21:16:02Z

Created page with '==Statement== The theorem states the following: For any <math>x > 0</math>, the number of fact about::twin primes less than or equal to <math>x</math> is bounded from above…'

New page

==Statement==

The theorem states the following:

For any <math>x > 0</math>, the number of [[fact about::twin primes]] less than or equal to <math>x</math> is bounded from above by:

<math>\frac{cx (\log \log x)^2}{(\log x)^2}</math>

where <math>c</math> is a positive constant independent of <math>x</math>.

As a corollary, we obtain that the sum, over all pairs of [[twin primes]], of the reciprocals of both members of the pair, is finite. Specifically, the following sum is finite:

<math>\left(\frac{1}{3} + \frac{1}{5}\right) + \left(\frac{1}{5} + \frac{1}{7}\right) + \left(\frac{1}{11} + \frac{1}{13}\right) + \dots</math>

The constant to which this sum converges is termed [[Brun's constant]]. No explicit finite upper bound on Brun's constant has been established.

The following are some easy corollaries:

# The sum of the reciprocals of all twin primes is finite. Note that this sum differs from the sum above by <math>1/5</math> since <math>1/5</math> appears twice in the above sum. In other words, the set of twin primes is a [[fact about::small set| ]][[proves property satisfaction of::small set]].
# The sum of the reciprocals of each of the ''lesser'' of the pairs of twin primes is finite.
# The sum of the reciprocals of each of the ''greater'' of the pairs of twin primes is finite.

Brun's theorem - Revision history

Vipul: Created page with '==Statement== The theorem states the following: For any x > 0, the number of fact about::twin primes less than or equal to x is bounded from above…'

Vipul: Created page with '==Statement== The theorem states the following: For any $x > 0$ , the number of fact about::twin primes less than or equal to $x$ is bounded from above…'