baseparities

Baseparities is a term used in discussions of number representation to denote parity-type invariants that depend on how an integer is written in a fixed base b ≥ 2. In this framework, a baseparity of an integer n is any quantity computed from the digits of n in base b and considered modulo 2. The concept encompasses several concrete measures, most commonly the parity of the digit sum and the parity of the number of nonzero digits.

Formally, if n is written in base b as n = sum_{i=0}^k d_i b^i with 0 ≤ d_i < b, two standard baseparities are:

- Digit-sum parity: (sum_i d_i) mod 2.

- Hamming parity: (number of i with d_i ≠ 0) mod 2.

These invariants are determined solely by the base-b digits of n and are stable under base-conversion operations that preserve digit patterns.

Key properties include a simple relation for odd bases. If b is odd, then n ≡ sum_i d_i (mod 2), so the digit-sum parity equals the parity of n itself. For even bases, the digit-sum parity need not match n’s parity, since factors of b^i with i ≥ 1 contribute even terms. The parity of nonzero digits, by contrast, behaves independently of this simple congruence and is influenced by the distribution of zero and nonzero digits in the expansion.

Baseparities have applications in digital-sum analysis, automata-theoretic characterizations of number representations, and in coding or cryptographic contexts where parity properties of digit structures are relevant. They also provide a compact lens for comparing numbers across bases and for studying carry phenomena in arithmetic.