Every integer greater than 5 is the sum of three primes?

[Math100]

Homework Statement

Prove that the Goldbach conjecture that every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of three primes.
[Hint: If $2n-2=p_{1}+p_{2}$, then $2n=p_{1}+p_{2}+2$ and $2n+1=p_{1}+p_{2}+3$.]

Relevant Equations

None.

Proof:

Let $a>5$ be an integer.
Now we consider two cases.
Case #1: Suppose $a$ is even.
Then $a=2n$ for $n\geq 3$.
Note that $a-2=2n-2=2(n-1)$,
so $a-2$ is even.
Applying Goldbach's conjecture produces:
$2n-2=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n=p_{1}+p_{2}+2$ is a sum of three primes.
Case #2: Suppose $a$ is odd.
Then $a=2n+1$ for $n\geq 3$.
Note that $a-3=2n-2=2(n-1)$,
so $a-3$ is also even.
Applying Goldbach's conjecture produces:
$a-3=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n+1=p_{1}+p_{2}+3$ is a sum of three primes.
Therefore, the Goldbach conjecture that every even integer greater than $2$ is the
sum of two primes is equivalent to the statement that every integer greater than $5$
is the sum of three primes.

 

[fresh_42]

Homework Statement:: Prove that the Goldbach conjecture that every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of three primes.
[Hint: If $2n-2=p_{1}+p_{2}$, then $2n=p_{1}+p_{2}+2$ and $2n+1=p_{1}+p_{2}+3$.]
Relevant Equations:: None.

Proof:

Let $a>5$ be an integer.
Now we consider two cases.
Case #1: Suppose $a$ is even.
Then $a=2n$ for $n\geq 3$.
Note that $a-2=2n-2=2(n-1)$,
so $a-2$ is even.
Applying Goldbach's conjecture produces:
$2n-2=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n=p_{1}+p_{2}+2$ is a sum of three primes.
Case #2: Suppose $a$ is odd.
Then $a=2n+1$ for $n\geq 3$.
Note that $a-3=2n-2=2(n-1)$,
so $a-3$ is also even.
Applying Goldbach's conjecture produces:
$a-3=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n+1=p_{1}+p_{2}+3$ is a sum of three primes.
Therefore, the Goldbach conjecture that every even integer greater than $2$ is the
sum of two primes is equivalent to the statement that every integer greater than $5$
is the sum of three primes.

You have shown that given Goldbach then every positive integer greater than five is a sum of three primes.

What is left to show is: If every integer $n>5$ can be written as $n=p_1+p_2+p_3$ the sum of three primes, then every even positive integer greater than two is the sum of two primes.

Equivalence means that one statement implies the other and the other way round.

(Forget about the greater than conditions. Just show it for integers great enough.)

 

[Math100]

Let $a>5$ be an integer.
Now we consider two cases.
Case #1: Suppose $a$ is even.
Then $a=2n$ for $n\geq 3$.
Note that $a-2=2n-2=2(n-1)$,
so $a-2$ is even.
Applying Goldbach's conjecture produces:
$2n-2=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n=p_{1}+p_{2}+2$ is a sum of three primes.
Case #2: Suppose $a$ is odd.
Then $a=2n+1$ for $n\geq 3$.
Note that $a-3=2n-2=2(n-1)$,
so $a-3$ is also even.
Applying Goldbach's conjecture produces:
$a-3=p_{1}+p_{2}$ as a sum of two primes $p_{1}$ and $p_{2}$.
Thus $2n+1=p_{1}+p_{2}+3$ is a sum of three primes.
Conversely, suppose every integer $a>5$ is the sum of three primes
such that $a=p_{1}+p_{2}+p_{3}$.
Let $a>2$ be an even integer.
Then $a+2=2n+2$ is even and $a+2>5$.
Thus $a+2=p_{1}+p_{2}+p_{3}$ is the sum of three primes $p_{1}, p_{2}$ and $p_{3}$.
Since $a+2$ is even,
it follows that at least one of $p_{1}, p_{2}$ and $p_{3}$ must be $2$.
Without loss of generality, assume $p_{3}=2$.
Then $a+2=p_{1}+p_{2}+2$.
Thus $a=p_{1}+p_{2}$ is a sum of two primes.
Therefore, the Goldbach conjecture that every even integer greater than $2$
is the sum of two primes is equivalent to the statement that every integer
greater than $5$ is the sum of three primes.

 

[fresh_42]

Looks good.

 

[Math100]

Thank you.