I Proving Goldbach's Conjecture for Math Proofs

[MevsEinstein]

TL;DR Summary

How many correct cases of a conjecture do you need so that it can be valid for your proof?

Lets say you were trying to prove a math statement when you realize that you can use a conjecture (say, Goldbach's conjecture) to finish the proof. If you don't have the time or the brains to prove it, how many cases of Goldbach's conjecture do you prove so that you can use it in your proof?

 

[lurflurf]

Your proof would be conditional. It would become valid once the conjecture is proven or false if the conjecture is disproven.

 

[Office_Shredder]

For what it's worth, there are tons of papers that do this.

https://en.m.wikipedia.org/wiki/Riemann_hypothesis
Contains a whole list of consequences that mathematicians proved to be true if the Riemann hypothesis is true, and are totally useless otherwise.

 

[mfb]

How many correct cases of a conjecture do you need so that it can be valid for your proof?

All of them. Testing an incomplete set of examples will never be a proof and we know of statements where the first counterexample is way beyond the reach of computers.
You can write "assuming the Goldbach conjecture is true, we prove that...", that's a standard procedure with well-known conjectures.

 

[jbriggs444]

All of them. Testing an incomplete set of examples will never be a proof and we know of statements where the first counterexample is way beyond the reach of computers.
You can write "assuming the Goldbach conjecture is true, we prove that...", that's a standard procedure with well-known conjectures.

Well, "all of the cases upon which the proof depends". If the proof depends on a something strictly weaker than the conjecture in question then a subset of cases could suffice.

If, for instance, a proof depended only on the correctness of Fermat's last theorem for n=5 then only cases where n=5 need be verified/proven.

 

[mfb]

In that case you wouldn't use or prove Fermat's theorem, you would prove a weaker statement. Which you still can't prove with examples because even the weaker statement is about an infinite set.

 

[Orodruin]

The two most common types of statements are either statements about existence or statements about non-existence*. If you want to prove that something exists, then a single example is enough. If you want to prove that something does not exist then you must exclude all candidates. You have either excluded all candidates or you have not. Coming with a set of examples that is not the full set of candidates is not a proof.

Goldbach’s conjecture is effectively a non-existence statement as it can be reformulated ”there is no even natural number greater than two that cannot be written as a sum of two primes”. Since there is an infinite number of such natural numbers, no finite set of examples that can be written as the sum of two primes is going to prove the conjecture.

However, the logical complement of the conjecture is a statement about existence ”there exists an even natural number larger than two that cannot be written as the sum of two primes”. A single example of such a number would be sufficient to prove this statement.

* Non-existence of an element with property A is the same as saying that all elements do not have property A. In the case if Goldbach’s conjecture, A is ”cannot be written as sum of two primes”. Not having property A is therefore ”can be written as a sum of two primes”.