Is there an algorithm help us to write each even number as sum of primes numbers? 1) Is there an algorithm help us to  write each even number as sum of primes numbers :
for example :$4=2+2$
$8=3+5$
where : $3,5,2$ are primes 
2) why we couldn't writing all odd numbers as sum of primes?
Late Edit: example look :$7=3+2+2$ 
I would be interest for any replies or any comments
 A: It is Goldbach's conjecture that every even number greater than 2 is the sum of two primes.  
Last year Harald Helfgott proved that every odd number greater than 5 is the sum of three primes.  
For the last decade, Tomás Oliveira e Silva has run distributed calculations to verify the Goldbach conjecture into the quintillions.
I have no idea about the algorithms.
A: Re: Original question, under the assumption that we are speaking of the sum of two primes...
Your first question is indeed Golbach's conjecture (addressing your comment below). It is an open question (as to whether every even number greater than $2$ can be written as the sum of two primes).
For your second question.
All prime numbers, with the exception of $2$, is odd. And odd + odd = even.
Some odd numbers can be written as the sum of two primes: $2+3 = 5$, $\;2+5 = 7, \;2+7 = 9,\;2+11 = 13,  2+ 17 = 19$. 
But in general, this won't work for all odd numbers.  
For example, $11 = 1+10 = 2+9 = 3+8 = 4+7 = 5+6$. All fail to be sums of two prime numbers. $17 = 1+16 = 2+15=3+14 = 4+13 = 5+12=6+11 = 7+10 = 8+9$. Again, $17$ (an odd number) fails to be the sum of two primes.
Adding two odd numbers (and so adding two odd primes) will always result in an even number.
A: The basic algorithm is simple: Given an even number n, iterate through the small primes (eventually up to n/2, though you're unlikely to need to go that far). For each $p$ check if $n-p$ is prime. If so you're done.
The expected work is about $0.5\log n$ primality tests with about $0.3\log^2n$ in the extreme cases. Of course it's not even known that you can find a solution, let alone by the $O(log^2n)$-th prime.
