Despite this is 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: $$x^5 + x + 1 = 0$$

What brings up this question is that "the radical real root for this specific example is impossible to be contained in a much wider range formula given as an independent following question for any polynomial degree as a trinomial polynomial (with all coefficients are zeros or (+/-) one (for only three terms),

however, that question which is an answer itself may not have been received well enough by some mathematicians mentioning that I'm not a professional mathematician which makes hard enough to convey the simplest idea to the professionals in this field

"Is this a NEW CORRECT One real approximated root in a rational form of the following $n$-th degree polynomial equation $$ x^n + x^m = 1,$$ where $(n > m)$ are two distinct positive integers, is given by the following series formula as an approximation where no real root ever exists exactly as we may imagine with our imaginative roots that are strictly associated or derived from that real root that never exists

$$ \begin{align} x &= 1-\frac{1}{n}+\sum_{k=2}^N \frac{(-1)^k \prod_{i=1}^{k-1}(km-in+1)}{k!n^k} \\ &= 1-\frac{1}{n}+\frac{2m-n+1}{2!n^2}- \frac{(3m-2n+1)(3m-n+1)}{3!n^3} +\frac{(4m-3n+1)(4m-2n+1)(4m-n+1)}{4!n^4} -\frac{(5m-n+1)(5m-2n+1)(5m-3n+1)(5m-4n+1)}{5!n^5} +\dotsb \end{align}$$

Where $N$ is a relatively small suitable chosen integer up to our practical needs and capability of making our real roots more sensible (only in rational form) for any other purpose since it is absolutely and perpetually impossible to make the sum of our terms go with no number like infinity since then our real root in our minds would simply become like a ratio of two no existing numbers (each with an uncountable number of digits) which isn't any number, especially that root also is impossible to be constructed exactly by any means

Note that in very rare cases, the above trinomial polynomial has real existing roots in constructible number form and strictly in their surd form (and never in their approximations in decimal or rational forms), when $(n = 2m)$, and $m$ is a power of two



For a reference to the more general solution of this Trinomial equation $$ax^n +bx^m +c = 0$$ click, OK, then click National then type in the search button "solution of equations by power series" at this link http://opac.nl.gov.jo/uhtbin/cgisirsi.exe/zJ7a8B3i0n/MAIN/263930002/123

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    $\begingroup$ I doubt there is a known "first" for the observation $x^5+x+1 = (x^2+x+1)(x^3-x^2+1)$. $\endgroup$ Commented Nov 10, 2015 at 14:13
  • $\begingroup$ The solution for the general trinomial was introduced by Lambert, and it's certainly not a new result. It was further developed by Euler in De Serie Lambertina. See my answer here: hsm.stackexchange.com/a/12490/9528 $\endgroup$ Commented Feb 21, 2021 at 1:20

2 Answers 2


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's not at all clear what this question is asking. The fact that the author posted the question 5-6 years ago, and only just replied to a comment, is also very odd. It probably doesn't deserve much consideration, but anyway, here are some notes.

(Adapted from my previous answer here)

In 1758, J. H. Lambert published a series solution for a real root of a general polynomial, in Observationes Variae in Mathesin Puram (Various observations on pure mathematics) in Act Helvetica Physico-Mathematico-Botanico-Medica Vol III. Considering as a special case the trinomial, $$ x^m + px = q $$ He obtains: $$ x = \frac{q}{p} - \frac{q^m}{p^{m+1}} + m \frac{q^{2m-1}}{p^{2m+1}} - m \frac{3m-1}{2!} \frac{q^{3m-2}}{p^{3m+1}} + m \frac{(4m-1)(4m-2)}{3!} \frac{q^{4m-3}}{p^{4m+1}} - ... $$ Or, in summation notation, $$S = \sum_{k=0}^\infty (-1)^k \frac{q^{k(m-1)+1}} {p^{km+1}} \frac{1}{k!} \prod_{j=0}^{k-2}(km-j)$$ It might seem like Lambert's solution is to a more specific problem, but in fact it has broad (enough) generality, with appropriate change of variables. I haven't checked, but it seems very plausible that the two series solutions are equivalent.

As for the fact that a 5th degree polynomial has five roots, Peter Roth published Arithmetica Philosophica (1608) which states that a polynomial of degree $n$ with real coefficients may have $n$ solutions (this information is available directly from the Wikipedia page for Fundamental theorem of algebra).


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