Rogers' Number Power Sum Problem, an extension of Fermat's last theorem.

I came up with an extension to Fermat’s last theorem.

I always thought that it was interesting that 

aⁿ + bⁿ = cⁿ where n was = 2, and 2 alone. 

3^2 + 4^2 = 5^2, is one of the solutions to Fermat's last theorem where n = 2, this is in fact Pythagoras's theorem for the diagonal for the hypotenuse in a right triangle. It has an infinite number of solutions. 

But then he took the exponent to 3 and up.  This always felt unbalanced to me. Like you need to add another term on the left each time you increment n.  My intuition feels like there has to be balance between the terms and the n value.

Roger’s Number Power Sum Problem:

For any positive integer n ≥ 2, consider the equation:

a₁ⁿ + a₂ⁿ + ... + aₙⁿ = zⁿ

Where a₁, a₂, ..., aₙ, and z are positive integers.

I tried it out with a simple example and it just worked. 

3^3 + 4^3 + 5^3 = 6^3
This does not fit the measurement of anything in a box or even a tesseract that I can figure out. 

The fact that this just worked the first  example simply by extending the same pattern is a very powerful sign to me that I am onto something.

That simple pattern breaks down at n=4 so far. My plan is to create a computer program to explore solutions. And to work on a general solution to this problem once I can see the patterns for solutions.  

And if this theorem only works for n=3 , then this is an amazing result too.  It would be very interesting to explore why it works for just one more example.  And if this is a general statement that shows there is a relationship between a power of n and n terms to the power of n, then that is also a very interesting result. 

I have a solution that proves this is not true for all positive numbers n>3, but the margin on this web page is too small to contain the answer.  (Just kidding. I don't plan on this being my last theorem.) 

And is 

aⁿ + bⁿ + cⁿ <> dⁿ for values of n bigger than 3? 

Is this a different proof than the proof that proved Fermat's last theorem? 

And is 

aⁿ + bⁿ + cⁿ + dⁿ <> eⁿ for values of n bigger than 4? 

Is this a different proof than the proof that proved the previous theorem, and that proved Fermat's last theorem?