Conjectures vs. Hypotheses

Date: 01/12/99 at 03:29:08
From: Ernest Lau
Subject: Number theory

What is the difference between the terms 'conjecture' and 'hypothesis'? 

For example, could the Riemann hypothesis be called the Riemann 
conjecture? What happens to a conjecture after it has been proven? Will 
it still be called a conjecture? For example, if the Taniyama-Shimura 
conjecture is proven in full, will it still be called that or something 
else?

Date: 01/12/99 at 09:59:05
From: Doctor Ezra
Subject: Re: Number theory

Dear Ernest,

Thanks for writing to Dr. Math. Let's take your questions in order:

A conjecture is a guess. Someone says, "I believe the following 
statement is true." Examples include the Twin Prime Conjecture (there 
are infinitely many primes p such that p+2 is also a prime), the 
Goldbach Conjecture (every even number can be written as a sum of two 
primes), and the Odd Perfect Numbers Conjecture (there are no odd 
perfect numbers). A conjecture is a statement for which someone thinks 
that there is evidence that the statement is true. The main thing about 
a conjecture is that there is no proof.

The word "hypothesis" has two meanings. The first is the "if" part of 
a theorem and the conclusion is the "then" part. (Example: In calculus, 
there is a theorem that states that if f is differentiable at a point 
c, then f is continuous at c. The hypothesis of this theorem is "f is 
differentiable at c" and the conclusion is "f is continuous at c.") In 
addition, mathematicians often use "hypothesis" as a substitute for 
"conjecture" - hence, the Riemann Hypothesis, which is really a 
conjecture. So, you're right: it ought to be the Riemann Conjecture!

Once a conjecture has been proven, it becomes a theorem. For example, 
the Four-Color Conjecture was a conjecture until 1976, when Appell and 
Haken constructed a proof. It is now the Four-Color Theorem. 

What all of this means is that Fermat's Last Theorem should have 
properly been called the Fermat Conjecture (but Fermat's Last Theorem 
had rather a nice ring to it...). It should now be known as the 
Fermat-Wiles Theorem; however, I suspect that people will refer to it 
as Wiles' proof of Fermat's Last Theorem.

Finally, as you said, once there's a full-blown proof of the
Taniyama-Shimura Conjecture, it will become a theorem. What people will 
call it is open to, well, conjecture!

Hope this helps.

- Doctor Ezra, The Math Forum
  http://mathforum.org/dr.math/