# Who was the first to show that this quintic equation has five radical roots?

Despite it being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: $$x^5 + x + 1 = 0$$

• I doubt there is a known "first" for the observation $x^5+x+1 = (x^2+x+1)(x^3-x^2+1)$. – Gerald Edgar Nov 10 '15 at 14:13

I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is:

$$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j).$$

It appears to be the first discovered general formula for real roots of the polynomial $x^n + x^m - 1 = 0$. It is not clear whether other radical roots can also be obtained using hypergeometric series.

*I would be interested in a discussion of mathematics forums and how they could be improved, and also how they could coordinate to address different kinds of questions.

I did find a "meta" question here about questions like: "Who first discovered X?". I couldn't participate in that discussion.

The reason for deleting BK's original answer (according to comment) is because it was not on topic of history of mathematics.

• It is so strange that new discoveries are painful to few mathematicians in power, I appreciate your interest in this discovery, wonder also about the silence of public mathematicians whom must support whatever is true, it could be not visible to them, I wonder!, at any case, It is only a very little result of mine from more general one, also documented in a book (1994), and had its own history from 1986, I can simply fill the word sites of mathematics of this only little formula, and the whole world can't simply refute it, it also contains many secrets I didn't mentioned yet, but any one can – Bassam Karzeddin May 11 '16 at 15:26
• This seems to be largely a for-profit forum, although I don't know for sure. I would recommend using Quora for asking questions like this in the future. – Gus Wiseman May 19 '16 at 0:41
• But, Quora also deleted the same formula without obvious reasons, it seems to me that it is determined in advance by some authority that they will never consider any thing in mathematics from any public domain unless they are the owners and regardless of its importance", and the purpose is also profit, other wise, a public domain "internet site" can simply produce something that top Journals can't, and this is would be the last nail in their existent purpose, which practically is going to vanish by the time due to huge and very fast world progress in communication technologies – Bassam Karzeddin May 19 '16 at 8:17