A note on unique representation of integers (using a sort-of-Latex notation): If, instead of powers of a base b (written as b^i), you have an increasing series of integers b_i, then the representation of an integer a in the form sum a_i b_i where 0 < a_i < (b_{i+1}/b_i, the representation is unique if and only if b_i divides b_{i+1} for every i.

For the regular case, since b_i = b^i, this is always true. A more interesting case is when b_i = i! (i factorial).

Anyway, I worked this out over 40 years ago and thought it might be of interest.

mjcohen

March 2, 2013@ 5:45 pmA note on unique representation of integers (using a sort-of-Latex notation): If, instead of powers of a base b (written as b^i), you have an increasing series of integers b_i, then the representation of an integer a in the form sum a_i b_i where 0 < a_i < (b_{i+1}/b_i, the representation is unique if and only if b_i divides b_{i+1} for every i.

For the regular case, since b_i = b^i, this is always true. A more interesting case is when b_i = i! (i factorial).

Anyway, I worked this out over 40 years ago and thought it might be of interest.