# Whether any even number can be written as sum of odd no of primes?

Whether any even number can be written as sum of odd number of primes?(3,5,7.. primes) I know that Goldbach's conjecture state that a even number can be written as sum of two primes.

D=A+B+C+...+n such that where no of elements in the equation is a odd number and where A,B,C are prime numbers and D is even number

whether the above one is true?

• $4$ cannot be written as the sum of an odd number of primes – Henry Dec 8 '12 at 16:02

## 4 Answers

All positive integers except $1$ and $4$ can be written as the sum of an odd number of primes.

$6=2+2+2$ can be written as the sum of an odd number of primes.

$7=2+2+3$ can be written as the sum of an odd number of primes.

$8=2+3+3$ can be written as the sum of an odd number of primes.

$9=3+3+3$ can be written as the sum of an odd number of primes.

By adding an even number of $2$s to these, any larger number can be written as the sum of an odd number of primes.

Looking at smaller numbers, $2$, $3$ and $5$ are equal to themselves and so are equal to the sum of an odd number of primes, since one is odd, while $1$ and $4$ cannot be written as the sum of an odd number of primes.

It is false. $4$ cannot be written this way.

We need a $2$. It turns out that for even $n \ge 8$, one $2$ is enough, the rest can be odd primes. This is easily verified for the first few even numbers $\ge 8$. After that, we can use induction, since if $n-6$ is a sum of an even number of odd primes, then so is $n$.

If Goldbach's conjecture is true, then you can do this by letting one of the primes is 2.D should be larger than 5.