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

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?

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

• PeroK
Staff Emeritus
Gold Member
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.

• dextercioby and PeroK
Mentor
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.

• dextercioby, jbriggs444 and Delta2
Homework Helper
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.

Mentor
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.

• jbriggs444
Staff Emeritus