Prime-matics

Improving the Error Term in the Sieve of Eratosthenes

The error term in Eratosthenes's sieve has remained unchanged for over a century. Some experts believe that bringing down the exponential in the error term would be nearly impossible due to the parity problem. However, avoiding the parity problem could lead to a possible solution. We have successfully found such a detour and significantly improved the error term.

There are infinitely many Twin Primes

We have developed a method for counting twin primes, which involves indirectly enumerating their midpoints using a simple elimination process. This alternative approach shows promise for finding a solution to the Twin Prime Conjecture through an asymptotic bound.

Grandma's Sieve

This video highlights that Grandma's Sieve offers at least two benefits over Eratosthenes's Sieve. Firstly, it eliminates duplications. Secondly, it employs a double summation instead of a single summation used in Eratosthenes's traditional sieve, providing more chances to adjust and improve the error term.

Onto the Twin Prime Conjecture

This note presents a more direct approach to verify the twin prime conjecture. We use a double summation to formulate the number of twin primes between z and x, where z grows with x. Our result shows that this value is always greater than zero, which confirms the twin prime conjecture.