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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?

lab bhattacharjee
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    Neither is a "confusion"; for the first, see this, and for the last, see this. – J. M. ain't a mathematician Aug 04 '12 at 07:37
  • I was asking what else are extant? – lab bhattacharjee Aug 04 '12 at 07:49
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    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. :) –  Aug 04 '12 at 09:01
  • 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 04 '12 at 10:02
  • Many mathematicians like Goldbach(http://mathworld.wolfram.com/GoldbachConjecture.html) considered one to be prime. I want know to the topics which ever produced some confusion ans it resolution(if any) as I've rephrased the question. – lab bhattacharjee Aug 04 '12 at 10:12
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    @Rahul Why do you think they are of very different kinds? They both boil down to definitions chosen for convenience. – Bill Dubuque Aug 04 '12 at 13:03
  • @Bill, aren't all definitions chosen for convenience? –  Aug 04 '12 at 20:46
  • @Rahul Sometimes. But why do you think they are "very different kinds"? – Bill Dubuque Aug 04 '12 at 21:58
  • @Bill: On second thought, I think you're right, and I withdraw my objection. I was thinking of the primality of 1 as an error, rather than merely a poor choice of definition. –  Aug 04 '12 at 23:17
  • @lab bhattacharjee: If your question is about mathematical terms which have historically had differing definitions, then that would be a fine question, and I think you should edit your question to say so clearly. I would suggest avoiding the word "confusion", as it's not obvious whether it means an error, a common misconception, or the existence of different conventions. (In particular, I don't think the fact that people disagree on whether 0 should be considered a natural number to be a "confusion"; neither side is confused here.) –  Aug 04 '12 at 23:19
  • @Rahul Yes, that was what puzzled me. Units are excluded from being prime or irreducible mainly for convenience, even though they do satisfy the respective divisibility properties. – Bill Dubuque Aug 04 '12 at 23:32

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

Gerry Myerson
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