About Two Goldbach’s Conjectures

elementary-number-theory

I've learned that

  1. "The Strong Goldbach's Conjecture" is that
    'All the even natural numbers greater than 2 can be written as a sum of two prime numbers.'
    And,
  2. "The Weak Goldbach's Conjecture" is that
    'All the natural numbers greater than 5 can be written as the sum of 3 prime numbers.'

But Sometimes people say that the weak conjecture is that
'All the odd natural numbers greater than 5 can be written as the sum of three prime numbers.'

Which one is correct? If it's the first one, then I think the weak conjecture is logically equivalent to the strong one. It's because of the following reasoning;

Strong$\implies$ Weak: If a natural number $n$ is greater than 5, then there are two cases;
i) $n$ is even: then we can write $n$ as $n=(n-2)+2 = p+q+2$, where $p, q$ are primes, by the strong conjecture($n-2>3$, so $n-2>2$ and also $n-2$ is even).
ii) $n$ is odd: then we can write $n$ as $n = (n-3)+3 = p+q+3$, where $p, q$ are primes, by the strong conjecture($n-3$ is even and $n-3>2$).
Weak$\implies$ Strong: All the even numbers can be written as the sum of three primes. But it's not possible that all three are odd primes. So there are at least one $2$. So if we subtract $2$ from $n$, we can conclude that all the even numbers greater than $2$ can be written as a sum of two primes.

As a result, I ask two things.

  1. Which one is correct version of "Goldbach's Weak Conjecture"?
  2. If the weak conjecture says about all the natural numbers, then why they aren't equivalent? I've heard that the weak conjecture was proven but strong is not. What is wrong with my reasoning?

Best Answer

As stated in the Origins section of Wikipedia's "Goldbach's conjecture" article,

On $7$ June $1742$, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture:

$\;\;\;\;\;$Every integer that can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units.

He then proposed a second conjecture in the margin of his letter:

$\;\;\;\;\;$Every integer greater than $2$ can be written as the sum of three primes.

He considered $1$ to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is:

$\;\;\;\;\;$Every integer greater than $5$ can be written as the sum of three primes.

It also later states

... Goldbach remarked his original (and not marginal) conjecture followed from the following statement

$\;\;\;\;\;$Every even integer greater than 2 can be written as the sum of two primes,

Thus, what you stated as what you learned to be the Weak Goldbach conjecture is actually basically just a restatement of Goldbach's strong conjecture that Goldbach made himself (apart from it starting at $2$ because he considered $1$ to be a prime), with it now known to be equivalent to what is now known at the Strong Goldbach conjecture, as you also determined & pointed out in your post.

The correct statement of Goldbach's weak conjecture is

$\;\;\;\;\;$Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum.)

which matches your second version.

Related Question