numbrina

Numbrina is a hypothetical mathematical construct used in introductory number theory discussions to explore how the digit-sum function behaves when a number is expressed in different positional bases. The term is not a standard entry in formal textbooks but appears in online expository writing and classroom exercises as a teaching aid to illustrate base-dependent properties of digits.

Definition and scope

For a finite set of bases B = {b1, b2, ..., bm} with each bi ≥ 2, a positive integer n is called a numbrina with respect to B if the digit-sum function s_b(n) is the same for every base b in B; that is, s_{b1}(n) = s_{b2}(n) = ... = s_{bm}(n). Here s_b(n) denotes the sum of the digits in the base-b representation of n. The concept is explicitly base-sensitive, since the same integer can have very different digit sums in different bases.

Examples and properties

Trivial examples include n = 0 and n = 1, for which s_b(n) is 0 or 1 in all bases. Nontrivial instances depend on the chosen base set B and typically require computation to verify. There is no general closed-form construction for numbrinas, and their existence is analyzed on a case-by-case basis.

Variants and usage

Some discussions extend the idea to allow near-constant digit sums, where s_b(n) can differ by only a small margin across bases, or to consider larger or infinite base families with specific asymptotic properties. In practice, numbrina serves educational purposes, fostering exploration of how base representation influences digit sums and encouraging hands-on calculation across multiple bases.

See also

Digit sum, Base representation, Number theory, Recreational mathematics.