## 1 + x + … + x^m = 1 + y + … + y^n

It appears that for   $N \; < \; 10^6$

$N \; = \; 1 \; + \; x \; + \; ... \; + \; x^m \; = \; 1 \; + \; y \; + \; ... \; + \; y^n$

holds only for

$31 \; = \; 1 \; + \; 5 \; + \; 5^2 \; = \; 1 \; + \; 2 \; + \; 2^2 \; + \; 2^3 \; + \; 2^4$

$8191 \; = \; 1 \; + \; 2 \; + \; 2^2 \; + \; ... \; + \; 2^{12} \; = \; 1 \; + \; 90 \; + \; 90^2$

in addition to evident solutions if   $x$   or   $y$   is negative.