Mathematics – General Mathematics
Scientific paper
2011-10-16
Mathematics
General Mathematics
33 pages
Scientific paper
The representation of even numbers as the sum of two primes and the distribution of primes in short intervals were investigated and a main theorem was given out and proved, which states: For every number $n$ greater than a positive number $n_{0}$, let $q$ be an odd prime number smaller than $\sqrt{2n}$ and $d=2n-q$, then there is always at least an odd number $d$ which does not contain any prime factor smaller than $\sqrt{2n}$ and must be an odd prime number greater than $2n-\sqrt{2n}$. Then it was proved that for every number $n$ greater than 1, there are always at least a pair of primes $p$ and $q$ which are symmetrical about the number $n$ so that even numbers greater than 2 can be expressed as the sum of two primes. Hence, the Goldbach's conjecture was proved. Also theorems of the distribution of primes in short intervals were given out and proved. By these theorems, the Legendre's conjecture, the Oppermann's conjecture, the Hanssner's conjecture, the Brocard's conjecture, the Andrica's conjecture, the Sierpinski's conjecture and the Sierpinski's conjecture of triangular numbers were proved and the Mills' constant can be determined. The representation of odd numbers as the sum of an odd prime number and an even semiprime was investigated and a main theorem was given out and proved, which states: For every number $n$ greater than a positive number $n_{0}$, let $q$ be an odd prime number smaller than $\sqrt{2n}$ and $d=2n+1-2q$, then there is always at least an odd number $d$ which does not contain any odd prime factor smaller than $\sqrt{2n}$ and must be a prime number greater than $2n+1-2\sqrt{2n}$. Then it was proved that for every number $n$ greater than 2, there are always at least a pair of primes $p$ and $q$ so that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. Hence, the Lemoine's conjecture was proved.
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