Computational Complexity Optimization in Mixed Binary Representation: The Q Algorithm

This paper introduces the Q Algorithm, a novel mixed binary numerical representation method that breaks away from the traditional pedagogical and historical paradigm of successive divisions and residue collection. Building upon the canonical decomposition N = m · 2k , where classical arithmetic defines and treats the factor m statically as an odd integer invariant, the Q Algorithm introduces a fundamental transformation by defining normalization through the relation m = q + 1. This reformulation allows the representation and processing of integers via discrete state tuples sn = (q, kn ), operating through direct jumps and cumulative exponent increments (∆kn ) to collapse intermediate structural zeros via the recurrence relation kn = kn−1 + ∆kn . Theoretically, we demonstrate that algorithmic complexity is optimized from a linear dependence on the total bit length O(n) toward a bound strictly governed by the Hamming weight O(H(N )). Through empirical case studies (such as the reduction from 14 to 7 operations for N = 8435), we show a substantial improvement in operational efficiency, opening new applications for optimized memory management in hardware and advanced cryptographic systems.