Proving Goldbach's conjecture by induction (or why it doesn't seem to work)

[mad mathematician]

Goldbach's claim: every even natural number $n>4$ can be written as a sum of two prime numbers.

Let's call this claim: G(n).
Let's try to solve it by induction.

For n=6=3+3, base check is correct.
Suppose G(n) is correct and let's try to prove that it implies G(n+2) is correct as well. (since if $n$ is even then n+1 is odd and n+2 is obviously even.

So $n+2=p_1+p_2+2$ where both p_i are prime numbers.

Now add $2$ to both sides and we get: $n+2=p_1+p_2+2$; now both the primes are obviously odd primes (I've excluded 4). I thought of saying that obviously $p_i+1$ are even numbers so if I assume for every even integer k<n we have also G(k), then obviously p_i+1 can be wrriten as a sum of two odd prime numbers. I want to prove the following claim: every sum of four odd primes can be reduced to a sum of two odd primes.

And this would allegdly prove Goldbach's conjecture, but it maybe that this claim is harder to prove...

 

[fresh_42]

And this would allegdly prove Goldbach's conjecture, but it maybe that this claim is harder to prove...

The sum of four odd primes is even. If Goldbach holds then it is the sum of two primes. Hence, it is a consequence of Goldbach. It does not prove Goldbach. Your "induction step" doubles the number of terms with every step. You need a limitation like Goldbach. Even the known limitations, like "every odd number greater than 1 is the sum of at most five prime numbers" (T.Tao , 2012) do not help here.

 

[mad mathematician]

The sum of four odd primes is even. If Goldbach holds then it is the sum of two primes. Hence, it is a consequence of Goldbach. It does not prove Goldbach. Your "induction step" doubles the number of terms with every step. You need a limitation like Goldbach. Even the known limitations, like "every odd number greater than 1 is the sum of at most five prime numbers" (T.Tao , 2012) do not help here.

What do you think, which tools in number can be handy in proving this hard conjecture?
I think I need to read some books on algebraic number theory.

 

[fresh_42]

What do you think, which tools in number can be handy in proving this hard conjecture?
I think I need to read some books on algebraic number theory.

I am no expert and Terry would certainly be far more competent to answer this question. He has proven many results like this so I would take a closer look at his paper(s). The two branches of mathematics that automatically come to mind in this context are class field theory and elliptic curves, both not exactly light fare.

There are often also analytical approaches in number theory like the one Nikolai Chudakov proved in 1937 that "almost all" even numbers are the sum of two prime numbers, that is, that the asymptotic density of the numbers representable in this way in the even numbers is 1. The goal here is to improve the limits of "almost all" or find lower bounds for possible counterexamples like Chen's theorem.

Prime numbers are strange. We can enumerate them, we know how often they occur (on average), yet, it took well over 300 years to prove Fermat's last theorem and its proof is 100 pages thick.