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What are the confusions & resolution(if any) , those have occurred in the history of the Theory of Numbers?

I have come across :

Is 1 a prime number:counter proof?

Is 0 a natural number: no general agreement about whether to include 0 in the set of natural numbers?

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Neither is a "confusion"; for the first, see this, and for the last, see this. – J. M. Aug 4 '12 at 7:37
I was asking what else are extant? – lab bhattacharjee Aug 4 '12 at 7:49
Your examples are of two very different kinds. The first has a clear answer and people who believe otherwise are simply mistaken. If that is what you want, see MathOverflow's examples of common false beliefs in mathematics. The second is a case where multiple different conventions are in use. If one specifies precisely what one means by the set of natural numbers, then there is no problem. The fact that you have clubbed these two different issues into the same category is, well, a confusion. :) – Rahul Aug 4 '12 at 9:01
The terms like natural or real number have come to use so that one does not have to specify their meaning every time, they are used, right? If I have to mention positive integers or non-negative integers whichever I want to mean, I would use them directly instead of mentioning confusing 'Natural' number, at all. – lab bhattacharjee Aug 4 '12 at 10:02
@Rahul Why do you think they are of very different kinds? They both boil down to definitions chosen for convenience. – Bill Dubuque Aug 4 '12 at 13:03

I think it took a while to figure out what should pass for the ring of integers in a number field.

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