You know, I have been playing with the Becerra's Theorem and it very interesting. For all of you that are not quite sure what this is about, I will explain:
Let n be a any integer base, k any power of n, and S(n) the sum of powers of n smaller than k, we can prove that:
S(n) × (n - 1) + 1 = k
I will give you an example: n = 4 (we choose base 2), and k = 64 (64 is one of the powers of 4). Then, S(n) = 4⁰ + 4¹ + 4² = 21. Note that that is where we stop because we need all the powers < 64. Then, we can see that:
21 × (4 - 1) + 1 = 64
This is very interesting, and its proof is not very known, because it involves Geometric series and so. They taught me this at school and I found this very surprising.