It is a known fact that powers of add upto any natural sum less than the next power of . This has been the basis for many practical problems with measurements, and allows us to count binary. The notation of this property is:
Then I realised, this could be extended to powers of , after the sum is multiplied by i.e.
This worked even for , when the sum was multiplied by
Thus, I thought I may generalize it:
, for all
Is this generalization correct, or is there a limit to ?
If there is a proof for this statement, can someone post it below?