Numbers: Not everything is magic

In my previous post - Number Magic, I scribbled that I found a pattern as follows...
- For any number n from 1 to 10 and any number k that satisfy the condition - k^n + k^(n+1) = (k+k)^n the next such condition is satisfied by (2k+1). But actually, there was an interesting solution. Lets put it in plain equation.

k^(n+1)+k^n = (2k)^n

=>k^n(k+1)=(2^n)*(k^n)

=>k=(2^n)-1

So, the number thats satisfies are 1, 3, 7, 15...1267650600228229401496703205375...

What is my learning here? - Get back to the basics... Or should I say to myself KISS - Keep it Simple, Stupid!!!