Talk:Number theory
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- Many questions in elementary number theory are exceptionally deep and require completely new approaches. Examples are
- The Goldbach conjecture concerning the expression of even numbers as sums of two primes,
- Catalan's conjecture regarding successive integer powers,
- The Twin Prime Conjecture about the infinitude of prime pairs, and
- The Collatz conjecture concerning a simple iteration.
What's that supposed to mean? Of the four conjectures, three haven't yet been proven. Mihailescu's proof of Catalan's conjecture has been said to use only elementary methods (and it has been said that the newer versions of the proof use no mathematical theory beyond what was available to Catalan himself).
As for the other three, I wouldn't rule out elementary proofs of any of them, and it is possible some of them cannot be proved at all. It's still not clear any of them would require completely new approaches.
Prumpf 15:26, 23 Nov 2003 (UTC)
I agree with the above. It doesn't make any sense to say they are exceptionally deep when they are very simple concepts, and it doesn't make any sense to claim that they require completely new approaches when we simply don't know.
Mysteronald 18:49, 5 Sep 2004 (UTC)
It does make some sense to say that they may lie deep or require new methods (which is not quite the same, but something closely related). Obviously simple assertion falls down on the test of being POV.
Charles Matthews 21:27, 5 Sep 2004 (UTC)
Sure. That does make some sense. This is a mathematical topic, and I think there's value in being precise in this instance. I've changed it, then.
Mysteronald 12:41, 6 Sep 2004 (UTC)
Combinatorial number theory should not be omitted! I hope someone could write description of this important branch of number theory.
Larry Hammick 07:44, 4 Apr 2005 (UTC)
Maybe the article should mention Diophantine approximation as a branch of number theory.
- But it does, under analytic number theory. I would regard combinatorial number theory as growing out of ANT, also, but I can see that not everyone would agree. Charles Matthews 10:57, 4 Apr 2005 (UTC)