sum powers with constant exponent

Question

I'd like to know if there is a solution to the following equation and how to solve it, thank you.

$$a^t + b^t + \cdots + z^t = Q$$

I want to find $t$, knowing $a,b,\cdots,z$ and $Q$

(Also is there a denomination to that sum)

Best Regards, FG

Answer 1

I do not think that you could get an analytical solution but numerically the problem is solvable quite efficiently. Considering that you look for the zero of

$$f(t)=\sum_{i=1}^n a_i^t- Q$$

it is better to consider instead the function $$g(x)=\log\left(\sum_{i=1}^{n} a_i^t\right)-\log(Q)$$ which is much better conditioned (it is almost a straight line).

An estimate of the zero could be obtained using a Taylor expansion of the logarithm at $t=0$ giving $$t_0=\frac {n(\log(Q) -\log(n)) } {\sum_{i=1}^{n} \log(a_i)}$$ but, since is equivalent to the first iteration of Newton method, we can simply use $t_0=0$.

Let us try with $n=12$, $a_i=p_i$ and $Q=123456789$. Using Newton method, the iterates would be $$\left( \begin{array}{cc} n & t_n \\ 0 & 0 \\ 1 & 6.538079375 \\ 2 & 4.994362680 \\ 3 & 4.986570543 \\ 4 & 4.986570264 \end{array} \right)$$ which the solution for ten significant figures.

If we repeat the problem using now $a_i=(-1)^i p_i$, the problem is more delicate starting at $t=0$ because of some indeterminations. Starting with $t=\frac 1 {10}$ would give as Newton iterates $$\left( \begin{array}{cc} n & t_n \\ 0 & 0.100000000 \\ 1 & 1.762045905 \\ 2 & 5.138386859 \\ 3 & 5.216134407 \\ 4 & 5.216145136 \\ 5 & 5.216145136 \end{array} \right)$$

Edit

Since in the question, instead of using $a_i$ as notation, you used $(a,b,c,\cdots,x,y,z)$, let us do it for $n=26$, $a_i=p_i$ and $Q=12345^{6789}$. The iterates will be $$\left( \begin{array}{cc} n & t_n \\ 0 & 0 \\ 1 & 18822.60277 \\ 2 & 13858.62238 \end{array} \right)$$ which the solution for ten significant figures.