QSS: Exact and Accelerated Evaluation of a Sum of Products of Integer Quotients
Авторы: Kim, Mark
Дата: Oct 2026

This paper considers the exact evaluation of the arithmetic sum of products of integer quotients G(N) for a positive integer parameter N.

We propose and investigate an implementation called QSS (Quotient-Split Summation), which combines direct evaluation of an initial segment with block grouping of indices by equal values of the first integer quotient. For each block, we prove that the second quotient takes at most two distinct consecutive values, allowing the block contribution to be computed analytically using a small, constant number of arithmetic operations.

In the standard arithmetic-operation model, the algorithm performs O(sqrt(N)) iterations and returns the exact integer result. In local Python measurements, QSS was approximately 27.57 to 2406.83 times faster than direct enumeration for the tested sizes N = 10^4 to 10^8. Compared with two alternative sublinear quotient-grouping implementations, the measured advantage was approximately 1.67 to 2.01 times due to reduced constant overhead.

These results apply to the specific software implementations and computer configuration tested and do not, by themselves, establish superiority over all existing algorithms.

DOI: 10.5281/zenodo.23286085

Recurrent Computation of Alternating Moments of Squared Binomial Coefficients
Авторы: Kim, Mark
Дата: Oct 2026

This paper addresses the computation of the alternating sum of squared binomial coefficients weighted by powers of integer indices. We describe an efficient exact method based on the derivatives of a finite hypergeometric polynomial and Stirling numbers of the second kind. A two-step recurrence relation for the derivatives at x = 1 is derived from the underlying differential equation. The proposed algorithm reduces the number of required arithmetic operations significantly compared to both direct summation and the previously known Kravchuk polynomial approach. Local benchmarking results demonstrate a substantial runtime advantage, achieving a speedup of up to 1062x in high-dimensional tests.

DOI: 10.5281/zenodo.23271148

Adaptive Precision Search for Integer Relations: The APS Algorithm
Авторы: Kim, Mark
Дата: Oct 2026

Finding integer relations is an important problem in computational mathematics. Given a set of real numbers, the task is to determine whether there exist small integer coefficients such that their linear combination equals zero. This paper presents APS (Adaptive Precision Search), an algorithm that uses an integer lattice embedding and LLL reduction to generate candidate integer relations.
Unlike a single, direct application of LLL, APS uses several levels of computational precision. At each level, an integer lattice is constructed, LLL reduction is performed to obtain a transformation matrix, and candidates are extracted from that matrix. The original real-valued residual is computed for each candidate. Candidates that remain stable as precision increases are preferred over accidental approximate relations.
In the experiments conducted, APS successfully recovered the prescribed relations in dimensions ranging from 4 to 30. At higher dimensions, a substantial runtime advantage over PSLQ was observed. In a dimension-30 test with a coefficient bound of 10, APS ran in approximately 0.027 seconds, whereas the PSLQ implementation used took approximately 6.41 seconds, achieving a speedup of approximately 258x.

DOI: 10.5281/zenodo.23270986