Talk:Euler's theorem

Math investigatory project

What is math investigatory project?? —Preceding unsigned comment added by 210.5.83.81 (talk • contribs) 07:19, 5 November 2004

Such a statement has now been removed from the article.--Leif edling (talk) 06:17, 18 May 2009 (UTC)

F+V=E+2?

I was taught that Euler's Theorem stated, that, in solids, the number of faces (F) plus the number of vertices (V)equals the number of edges (E) plus two. Is that not Euler's Theorem? —Preceding unsigned comment added by 207.225.245.182 (talk) 00:33, 29 May 2009 (UTC)

There are many mathematical results named after Euler. The one you mention is sometimes called Euler's formula, and is generalised in the Euler characteristic. Gandalf61 (talk) 09:52, 29 May 2009 (UTC)

Statement

This theorem is stated as an implication: if a and n are coprime, then we know that the value $a^{\varphi(n)} \bmod n$ is 1. That's the way the theorem is stated here and in most textbooks (I just checked Fundamental number theory with applications by Richard A. Mollin, page 93). This is the non-trivial result.

The converse is true, but it's so trivial that it does not have to be stated. An equivalent form of the converse is: if a and n are not coprime, then we know that the value $a^{\varphi(n)} \bmod n$ is not 1. This is obvious: Let p be any prime that divides both a and n, then for all positive integers x we know that p divides $a^x \bmod n$, hence this value is not 1.

Please keep the statement as commonly used. The converse is barely worth a note somewhere in the article. Misof (talk) 08:30, 18 March 2011 (UTC)

I completely agree. The article must give a statement of the theorem as it appears in most reliable sources, which is as a one-way implication. I don't mind whether we also mention the converse or not - as Misof says, it is trivially true. But we must not change the statement of the original theorem. Gandalf61 (talk) 09:15, 18 March 2011 (UTC)