doublingtwo

Doublingtwo is a hypothetical mathematical operation used to illustrate properties of binary representations. The name combines “doubling” with “base two,” reflecting that the transformation operates on binary strings by duplicating them.

Definition: For any positive integer n, let k be the number of bits in its binary form (k = floor(log2 n) + 1). The doublingtwo transform D maps n to a new integer whose binary representation consists of the binary form of n followed by itself. Equivalently, D(n) = n*(2^k + 1).

Examples: n=5 (binary 101) yields D(5) = 101101_2 = 45. n=4 (binary 100) yields D(4) = 100100_2 = 36. The image consists of binary strings of the form XX, where X is the binary representation of n.

Properties: D is strictly increasing and injective, since equal outputs would imply identical inputs and equal bit-length. The function is not surjective onto the natural numbers because only numbers whose binary expansion is a double of a binary string (XX) are in its image. The doublingtwo image consists exactly of numbers whose binary representation can be divided into two equal halves.

Applications and context: Doublingtwo is primarily of theoretical interest in studies of binary encoding and string concatenation; it can serve in puzzle design, encoding exercises, or as a simple example of a base-two self-similarity operation. It has no widely recognized practical applications and is not a standard mathematical operator.