Solving $1^x+2^x+3^x=0$ equations... Is it possible to solve for x this kind of equation? Since 1,2,3 are not multiples to each other I see a priori no possibility.
$$1^x+2^x+3^x=0; x?$$
Computing this on Wolfram Alpha, for example,  returns two complex solutions for $x$. Wolfram Alpha usually returns a formal and algebraical expression to represent the solution, but this time no formal one is returned, instead, just a long, infinite and complex number with seems to be the result of an iterating computation rather than a formal way to solve it.
You can check the results with Newton-Rhapson's method in my GeoGebra's plot.

It certainly can't have any real value solutions since,
$$1^x+2^x+3^x=0$$
$$1+2^x+3^x=0$$
$$2^x+3^x=-1$$
And $a^x>0$ for all real $x$ and all real $a>0$.
So it makes sense that if you want a solution, the search needs the extended number system of complex numbers.
If you ask Wolfram Alpha to solve
$$2^x+3^x=-1$$
it will give you the numerical solutions
$$x=-0.454397 \pm 3.59817i$$
If you want to try and obtain these by hand you could try applying the numerical Newton Raphson method to the equation; it works with complex numbers as well as the reals, but the arithmetic may be daunting especially if you are interested in solutions to
$$\sum_{k=1}^{n} k^x=0$$
To explore this properly would need software that can apply the Newton Raphson method in complex numbers and show you what it's doing.
A quick google search makes me think Maxima may do it for free : http://maxima.sourceforge.net/
or GNU Octave : http://www.gnu.org/software/octave/
although I have no experience of either, or even if they can cope with complex numbers.
The irony is, of course, that you ideally want to start near a solution. I see this is becoming an interesting project; be good to graph the modulus of the complex function for small values of k, as often done for Zeta function.
Update:
There's a free interactive Complex Grapher embedded within a webpage which I've used to look at the zero at -0.45+3.59i which is shown at the black centre of the coloured circle.

The webpage is here : https://www.complexgrapher.com/
