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.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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