digitcompatibility

Digitcompatibility is a term used in number theory and computer science to describe a digit-level relationship between two integers or sequences of digits. Under a fixed base b, write two nonnegative integers as digit strings a = a_{n-1}...a_0 and c = c_{n-1}...c_0 with digits a_i, c_i in {0,...,b-1}. The two numbers are said to be digit-compatible with respect to a binary relation R on the digit set if for every position i, the pair (a_i, c_i) belongs to R.

This framework allows a variety of per-digit constraints to be formalized. Common forms include:

- Additive carryless compatibility, where R consists of all (x,y) with x + y < b, meaning the two digits sum without producing a carry at that position.

- Multiplicative compatibility, where, for a given target per-digit result d_i, the digits satisfy x * y ≡ d_i mod b.

- Equivalence-type compatibility, where x and y belong to a predefined subset or satisfy a congruence relation, such as x ≡ y mod m for some m.

Properties of digitcompatibility typically rely on independence across positions: the relation at each digit is applied identically to all positions, and the overall compatibility is the conjunction of per-digit conditions. If R is symmetric or transitive, analogous properties may carry over to the global relation on whole numbers.

Applications and relevance include error-detection schemes in digitwise transmission, design of carryless arithmetic circuits, cryptographic constructions that use per-digit constraints, and puzzle or code design where digit-wise compatibility governs valid pairs.

Note: digitcompatibility is not a standard, widely adopted term; the article presents a general, formal framework for digit-level relations between numbers. See also carryless arithmetic, base-b representations, and digitwise operations.