A 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.

 

[martinbn]

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.

I doubt it, of course one never knows, but Goldbach is an additive number theory problem, and CFT and EC don't seem to handle such questions.

 

[fresh_42]

I doubt it, of course one never knows, but Goldbach is an additive number theory problem, and CFT and EC don't seem to handle such questions.

CFT and EC are my automatic associations when it comes to primes, having FLT in mind. And, e.g.

In 1947, Alfréd Rényi proved that a constant K exists such that every even number is the sum of a prime number and a number with at most K prime factors.

shows that multiplications aren't excluded. Just prove K=1.

 

[martinbn]

CFT and EC are my automatic associations when it comes to primes, having FLT in mind. And, e.g.

Yes, but Goldbach has very different nature.

shows that multiplications aren't excluded. Just prove K=1.

How did he prove it? Was CFT or EC used?

 

[fresh_42]

How did he prove it? Was CFT or EC used?

Unfortunately, I do not speak Russian so it makes no sense for me to have a look at it. If you do then here is the citation: Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 12: 57–78.
Translated title: "On the representation of an even number as the sum of a prime and an almost prime"

I simply do not dare to rule out major branches of number theory if I have no idea about possible approaches.

A look at Tao's paper ...

... our additional techniques, which may be of interest for other Goldbach-type problems, include the use of smoothed exponential sums and optimisation of the Vaughan identity parameters to save or reduce some logarithmic losses ...

... immediately leads to number-theoretical functions like the Mangoldt function and Dirichlet L-functions and we find ourselves in the realm of ERH. I am sure that the early proofs of FLT for $n=3,4$ looked very different from Wiles' final proof, too. Whoever will prove Goldbach will certainly be an expert in all possible methods of number theory, including CLF and EC.

Edit: I agree, thinking about the close relationship between Rényi and Erdös, that it's more likely that he used analytical methods and not algebraic ones. But I also think that Goldbach is so hard that you cannot afford to not know all you can get from number theory.

 

[martinbn]

I don't speak Russian either, but I can read it and I looked at the paper. It uses standard analytic number theory techniques but no CFT or EC.

What Tao describes are techniques that you will not find in any textbook on CFT or EC.

Of course I cannot exlude the possibility that CFT or EC will be needed for Goldbach, all i am saying is that they not the obvious areas to suggest.

 

[fresh_42]

I agree. Rényi and Tao are more on the book's analytical page than the algebraic. However, the analytical approach has not succeeded so far. It cannot be ruled out that even new mathematics has to be invented, hopefully of Wiles' kind rather than Mochizuki's.

I was surprised that there is a proof for algebraic function fields:
https://arxiv.org/abs/1808.04001

 

[martinbn]

I agree. Rényi and Tao are more on the book's analytical page than the algebraic. However, the analytical approach has not succeeded so far. It cannot be ruled out that even new mathematics has to be invented, hopefully of Wiles' kind rather than Mochizuki's.

Of course, one never knows.

I was surprised that there is a proof for algebraic function fields:
https://arxiv.org/abs/1808.04001

The function field case is always easier than the number field case.