Prove that the Goldbach conjecture that every even integer....

[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 $n$ be an integer.
Then $2n=p_{1}+p_{2}$ for $n\geq 2$ where $p_{1}$ and $p_{2}$ are primes.
Suppose $n=k-1$ for $k\geq 3$.
Then $2(k-1)=p_{1}+p_{2}$
$2k-2=p_{1}+p_{2}$
$2k=p_{1}+p_{2}+2$.
Thus $2k+1=p_{1}+p_{2}+3$.
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.

 

[nuuskur]

1. Please use more concise titles.
2. It's an if and only if statement. How does the statement "every integer greater than $5$ is the sum of three primes" imply Goldbach according to your work?

 

[PeroK]

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 $n$ be an integer.
Then $2n=p_{1}+p_{2}$ for $n\geq 2$ where $p_{1}$ and $p_{2}$ are primes.
Suppose $n=k-1$ for $k\geq 3$.
Then $2(k-1)=p_{1}+p_{2}$
$2k-2=p_{1}+p_{2}$
$2k=p_{1}+p_{2}+2$.
Thus $2k+1=p_{1}+p_{2}+3$.
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.

Your problem is that you can prove anything. You simply put together a few lines and declare the result proved. These questions that tell you the answer are no good. You are not learning much from them.

These questions you are doing are only useful to students who can check their own work and spot their own errors - which you never do.

You need to find questions where you are asked to prove something if it is true or find a counterexample if it is false. That means you don't know in advance whether the statement you are given is true of not.

You can't possibly progress if after all the problems you've posted on here you are still submitting work like this. It's substandard, IMO. In any case, you need to learn to check your own work and find your own errors.

 

[nuuskur]

Practice has shown that you are stacking some, perhaps somewhat related, statements in a proof attempt and then declare the statement proved. On the other hand, when pried for details, your responses are often nonsensical.

Spoiler

As an example, I asked you in another thread why it's important for $a$ to be nonnegative for the claim that involves $\sqrt[n]{a}$. Your response to it is circular.

The reason is the quantity $\sqrt[n]{a}$ is not well defined for negative $a$.

Furthermore, the claim in this problem is an equivalence, which means one needs to prove two implications. There is no mention of this anywhere in your work. Further criticisms

1. You make no mention of assumptions.
2. You don't quantify statements.
3. You focus a lot on obvious details, while not recognising (or purposefully dodging?) central arguments.

You keep repeating the same errors, you don't acknowledge the criticism you are given. This is extremely discouraging. Vaas Montenegro (from Farcry 3) might have some comments about your behaviour

 

[pbuk]

All you have done is restated the hint (if $2n-2=p_{1}+p_{2}$, then $2n=p_{1}+p_{2}+2$ and $2n+1=p_{1}+p_{2}+3$).

The key to this proof is understanding why the trivial facts in the hint are relevant.

Spoiler: A hint to the hint

If $p_{1}+p_{2}+p_{3}$ is even what does this say about $p_{1}, p_{2} \text{ and } p_{3}$?

As others have pointed out the problem is not that you didn't work out how to do this proof, it is that you convinced yourself that what you had done constituted a proof.

 

[berkeman]

Thread is closed for Moderation...

 

[berkeman]

Thread will remain closed. OP -- please check all of your PMs. Thanks.