[
    {
        "title": "Adaptive Precision Search for Integer Relations: The APS Algorithm",
        "authors": "Kim, Mark",
        "description": "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.\r\nUnlike 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.\r\nIn 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.",
        "url": "https://zenodo.org/records/23270986",
        "doi": "10.5281/zenodo.23270986",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Recurrent Computation of Alternating Moments of Squared Binomial Coefficients",
        "authors": "Kim, Mark",
        "description": "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.",
        "url": "https://zenodo.org/records/23271148",
        "doi": "10.5281/zenodo.23271148",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "QSS: Exact and Accelerated Evaluation of a Sum of Products of Integer Quotients",
        "authors": "Kim, Mark",
        "description": "This paper considers the exact evaluation of the arithmetic sum of products of integer quotients G(N) for a positive integer parameter N. \r\n\r\nWe 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. \r\n\r\nIn 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. \r\n\r\nThese results apply to the specific software implementations and computer configuration tested and do not, by themselves, establish superiority over all existing algorithms.",
        "url": "https://zenodo.org/records/23286085",
        "doi": "10.5281/zenodo.23286085",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Accelerating the Computation of Hexadecimal Digits of Pi Using Bellard's Formula by Grouping Fractions",
        "authors": "Kim, Mark",
        "description": "This paper addresses the acceleration of computing arbitrary hexadecimal digits of pi using Bellard's formula. We propose a computational modification in which seven fractions with linear denominators are combined into a single rational function with a common denominator. This replacement reduces the number of modular exponentiations from seven to one for each term of the series. The mathematical basis of the transformation, the digit extraction algorithm, and integer arithmetic constraints are described. Local benchmarks in Python 3.12.6 demonstrate a runtime advantage of 1.93x to 2.72x over the standard implementation. Furthermore, the applicability of this fraction grouping method is investigated and successfully verified against Smith's formula.",
        "url": "https://zenodo.org/records/23348803",
        "doi": "10.5281/zenodo.23348803",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Sublinear O(N^(1/3)) Computation of Quotient Collisions via Interval Grouping",
        "authors": "Kim, Mark",
        "description": "This paper addresses the problem of exactly computing the number of quotient collisions E(N), defined as the number of index pairs i < j for which the integer parts of N/i and N/j coincide, as well as computing the exact distribution of quotient block sizes. The standard approach requires iterating over approximately sqrt(N) blocks. We propose a novel sublinear algorithm based on two key structural observations: the block size c_w takes one of two adjacent values m or m+1, where m = floor(N/(w(w+1))), and the sum of block sizes over an interval of constant m can be evaluated telescopically using only two values of the integer part. This allows evaluating the contribution of an entire interval in O(1) arithmetic operations, reducing the number of processed intervals to at most 2*ceil(N^(1/3)). We present the mathematical foundations, the algorithmic implementation, complexity bounds, and verification procedures. Empirical benchmarks in Python demonstrate a practical speedup ranging from 1.21x to 51.33x for N between 10^6 and 10^16, with the performance gain increasing asymptotically with N.",
        "url": "https://zenodo.org/records/23385618",
        "doi": "10.5281/zenodo.23385618",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Exact Counting of Right Triangles with Vertices at the Nodes of an n x n Grid: Algorithms of Complexity O(n^(3/2+eps)) and O(n log^2 n).",
        "authors": "Kim, Mark",
        "description": "This paper considers the exact computation of the number R(n) of right triangles with vertices at the nodes of an n x n grid (sequence A077435 in the OEIS). A formula is obtained in the form of a fourfold sum, R(n) = 4S^2 + 4W, over p,q,g,mu >= 1 with gcd(p,q)=1. The summand is symmetric under the exchange of the pairs (g,mu) and (p,q); from this an algorithm of complexity O(n^(3/2+eps)) is derived. Next, the coprimality condition is removed by means of the Mobius function, the sum is split into layers, and the remaining two-dimensional sums over polygons are computed by a kernel of floor sums with moments in O(log n); the overall complexity is O(n log^2 n) arithmetic operations. Both Python implementations coincided with direct enumeration for n <= 7, with all OEIS values for n <= 35, and with the OEIS b-file up to n=10,000. In the measurements, the near-linear algorithm turned out to be faster at n=12,800 (9.35 s versus 11.55 s), yielding the previously unknown exact value of R(12800).",
        "url": "https://zenodo.org/records/23439957",
        "doi": "10.5281/zenodo.23439957",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Sublinear Exact Counting of Acute, Right and Obtuse Integer Triangles with Given Perimeter",
        "authors": "Kim, Mark",
        "description": "This paper considers the exact number of integer-sided triangles with perimeter p that are acute, right or obtuse. The formulas listed in the OEIS for these counts use double sums, requiring O(p^2) operations, and a single sum over the shortest side gives O(p). It is shown that, with u=p-a and v=p-b for the two shorter sides a <= b, the type of the triangle is decided by the sign of 2uv-p^2, so the acute triangles are the lattice points of a polygon lying under the hyperbola uv=p^2/2. The count reduces to one hyperbolic sum over a band of ratio sqrt(2), a few closed-form sums, and the number of divisor pairs of p^2/2 for the right triangles. With a hull-walk kernel for hyperbolic sums, this gives O(p^(2/3)) operations. The Python 3.12 implementation coincided with direct enumeration and baseline evaluations. At p=10^9+7 the linear method took 319.6 s, while the new method took only 3.2 s (100x speedup), enabling exact partition counts up to p=10^11+7.",
        "url": "https://zenodo.org/records/23473044",
        "doi": "10.5281/zenodo.23473044",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Exact Counting of Pythagorean Triangles with Bounded Legs: Sublinear Algorithms and New Terms of OEIS A239581, A239744, A239786",
        "authors": "Kim, Mark",
        "description": "This paper provides an exact sublinear computation of the number of Pythagorean triangles with legs x < y <= N, considering both primitive triangles Q(N) and all triangles T(N). The problem is connected to the sequences A239581, A239744, and A239786 in the OEIS, where terms were previously known only up to N = 10^11. We demonstrate that the plane of Euclid pairs (m,n) is partitioned by the ray m = (1+sqrt(2))n into two symmetric hyperbolic regions under different coordinates: mn <= Y/2 and uv <= Y (with u=m-n, v=m+n). The boundary is evaluated exactly using Beatty floor sums with a 40-digit approximation of sqrt(2), and the coprimality condition is handled via Möbius inversion. The resulting algorithm achieves O(sqrt(N) log N) for Q(N) and O(N^(2/3)) for T(N). We successfully reproduce all 11 known OEIS terms and extend the sequences by computing new terms up to N = 10^17 for primitive triangles (a(17) = 17860363013590019) and N = 10^13 for all triangles.",
        "url": "https://zenodo.org/records/23518201",
        "doi": "10.5281/zenodo.23518201",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    },
    {
        "title": "Exact Counting of Integer Triangles with a 120-Degree Angle by Perimeter: Wedge-Shaped Hyperbolic Sums and Möbius Inversion",
        "authors": "Kim, Mark",
        "description": "This paper considers the exact computation of the number of primitive Q(N) and all T(N) integer-sided triangles with a 120-degree angle and perimeter at most N. Standard methods scale linearly with O(N) complexity. We show that primitive triangles correspond one-to-one to pairs (x,y) satisfying 3x < 2y < 4x, gcd(x,y)=1, and 3 does not divide y, where the perimeter equals the product xy. The problem is thus reduced to evaluating a wedge-shaped hyperbolic sum under xy <= Z. By splitting the summation at a threshold of Z^(1/3) and removing the coprimality condition via Möbius inversion over integers coprime to 3, we obtain an O(sqrt(N) log N) algorithm for Q(N) and an O(N^(2/3)) algorithm for T(N). We derive the exact asymptotic density for primitive partitions as 9*ln(4/3)/(4*pi^2) * N. Python 3.12 implementations demonstrate substantial speedups against linear baselines, enabling the calculation of exact counts up to N = 10^16 for Q(N) and N = 10^13 for T(N).",
        "url": "https://zenodo.org/records/23561814",
        "doi": "10.5281/zenodo.23518201",
        "date": "Oct 2026",
        "type": "",
        "status": ""
    }
]