We show that the nth digit of the base-b representation of any quadratic irrational a is a finite-state function of the Ostrowski a-representation of bn, and hence can be computed by a finite automaton. We use a satisfiability (SAT) solver to prove, for some quadratic irrationals, that the automata we construct are both minimal and unique. For other quadratic irrationals, the SAT solver is able to find smaller automata computing the digits of the irrational up to a high precision. We conjecture in these cases that the automata found do indeed compute all digits of the irrational correctly. We give a heuristic argument for this conjecture, though we leave this as an open question.

Ever since E. T. Parker constructed an orthogonal pair of 10×10 Latin squares in 1959, an orthogonal triple of 10×10 Latin squares has been one of the most sought-after combinatorial designs. Despite extensive work, the existence of such an orthogonal triple remains an open problem, though some negative results are known. In 1999, W. Myrvold derived some highly restrictive constraints in the special case in which one of the Latin squares in the triple contains a 4 × 4 Latin subsquare. In particular, Myrvold showed there were twenty-eight possible cases for an orthogonal pair in such a triple, twenty of which were removed from consideration. We implement a computational approach that quickly verifies all of Myrvold's nonexistence results and in the remaining eight cases finds explicit examples of orthogonal pairs—thus explaining for the first time why Myrvold's approach left eight cases unsolved. As a consequence, the eight remaining cases cannot be removed by a strategy of focusing on the existence of an orthogonal pair; the third square in the triple must necessarily be considered as well. Our approach uses a Boolean satisfiability (SAT) solver to derive the nonexistence of twenty of the orthogonal pair types and find explicit examples of orthogonal pairs in the eight remaining cases. To reduce the existence problem into Boolean logic we use a duality between the concepts of transversal representation and orthogonal pair and we provide a formulation of this duality in terms of a composition operation on Latin squares. Using our SAT encoding, we find transversal representations (and equivalently orthogonal pairs) in the remaining eight cases in under two hours of computing on a large computing cluster.

Latin squares are n× n matrices containing n symbols, where each symbol appears exactly once in each row and column. They were studied by Euler, later popularized through Sudoku, and remain a rich source of difficult combinatorial search problems. Two Latin squares are orthogonal mates if, when overlaid, no ordered pair of symbols repeats. Pairs of orthogonal Latin squares exist for every order except 2 and 6, but finding orthogonal Latin squares computationally can be challenging. Satisfiability (SAT) solvers are strong at combinatorial search and have been used to resolve a number of various kinds of orthogonal Latin square problems. On the other hand, SAT solvers lack domain knowledge about Latin squares, such as the Euler-Parker algorithm for orthogonal mate construction. In this paper, we propose a hybrid method combining a SAT solver with the Euler-Parker algorithm (implemented using a Diophantine system solver) and show that the resulting solver is effective at finding certain kinds of orthogonal Latin squares. For example, certain pairs of 10×10 orthogonal Latin squares whose existence was unknown for over 25 years were recently found by Bright, Keita, and Stevens using a SAT solver. The hardest cases could not be solved by the SAT solver CaDiCaL within seven days, but CaDiCaL augmented with an external Euler-Parker algorithm solves these cases in a median of around 5,100 seconds.

A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in Williamson's construction of quaternion-type Hadamard matrices. Using this correspondence, we devise an enumeration algorithm that is significantly faster than previously used algorithms and does not require the sequences to be symmetric. We implement our algorithm and use it to enumerate all circulant and possibly non-symmetric Williamson-type matrices of orders up to 21; previously, the largest order exhaustively enumerated was 13. We prove that when the blocks of a quaternion-type Hadamard matrix are circulant, the blocks are necessarily pairwise amicable. This dramatically improves the filtering power of our algorithm: in order 20, the number of block pairs needing consideration is reduced by a factor of over 25,000. We use our results to construct quaternionic Hadamard matrices of interest in quantum communication and prove they are not equivalent to those constructed by other means. We also study the properties of quaternionic Hadamard matrices analytically, and demonstrate the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. These results indicate a richer set of properties and suggest an abundance of quaternionic Hadamard matrices for sufficiently large orders.

We present a search for small Kochen-Specker (KS) sets in dimension 3, specifically targeting extensions of the 13-ray Yu-Oh set, which has been proven to be the minimal witness to state-independent contextuality. To enable this search, we introduce a novel SAT-based orderly generation framework integrating recursive canonical labeling (RCL) with the graph isomorphism tool NAUTY. We demonstrate that previous SAT approaches relying on lexicographical canonicity suffer from exponential scaling on canonical graphs. This limitation renders them intractable on the large instances (25 to 33 vertices) encountered in our search, whereas our RCL check maintains consistent millisecond-level performance, effectively eliminating the bottleneck. Overcoming this bottleneck allows us to perform the first exhaustive enumeration of all KS sets with up to 33 rays containing the complete 25-ray state-independent contextuality (SI-C) set obtained by rigid extensions of the Yu-Oh set in 1,641 CPU hours. We found and verified that the 33-ray set discovered by Schütte is the smallest three-dimensional KS set containing the complete 25-ray SI-C set. All non-existence results are backed by independently verifiable proof certificates via an extension of the DRAT proof format.

We investigate the Gerver-Ramsey collinearity problem of determining the maximum number of points in a north-east lattice path without k collinear points. Using a satisfiability solver, up to isomorphism we enumerate all north-east lattice paths avoiding k collinear points for k <= 6. We also find a north-east lattice path avoiding k = 7 collinear points with 327 steps, improving on the previous best length of 260 steps found by Shallit. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

A k-net(n) is a combinatorial design equivalent to k - 2 mutually orthogonal Latin squares of order n. A relation in a net is a linear dependency over F-2 in the incidence matrix of the net. A computational enumeration of all orthogonal pairs of Latin squares of order 10 whose corresponding nets have at least two nontrivial relations was achieved by Delisle in 2010 and verified by an independent search of Myrvold. In this paper, we confirm the correctness of their exhaustive enumerations with a satisfiability (SAT) solver approach instead of using custom-written backtracking code. Performing the enumeration using a SAT solver has at least three advantages. First, it reduces the amount of trust necessary, as SAT solvers produce independently-verifiable certificates that their enumerations are complete. These certificates can be checked by formal proof verifiers that are relatively simple pieces of software, and therefore easier to trust. Second, it is typically more straightforward and less error-prone to use a SAT solver over writing search code. Third, it can be more efficient to use a SAT-based approach, as SAT solvers are highly optimized pieces of software incorporating backtracking-with-learning for improving the efficiency of the backtracking search. For example, the SAT solver completely enumerates all orthogonal pairs of Latin squares of order 10 with two nontrivial relations in under 2 h on a desktop machine, while Delisle's 2010 search used 11,700 CPU hours. Although computer hardware was slower in 2010, this alone cannot explain the improvement in the efficiency of our SAT-based search.

Given a linear equation E of the form ax + by = cz where a, b, c are positive integers, the k-colour Rado number R-k(E) is the smallest positive integer n, if it exists, such that every k-colouring of the positive integers {1, 2, . . . , n} contains a monochromatic solution to E. In this paper, we consider k = 3 and the linear equations ax + by = bz and ax + ay = bz. Using SAT solvers, we compute a number of previously unknown Rado numbers corresponding to these equations. We prove new general bounds on Rado numbers inspired by the satisfying assignments discovered by the SAT solver. Our proofs require extensive case-based analyses that are difficult to check for correctness by hand, so we automate checking the correctness of our proofs via an approach which makes use of a new tool we developed with support for operations on symbolically-defined sets-e.g., unions or intersections of sets of the form {f(1), f(2), . . . , f(a)} where.. is a symbolic variable and f is a function possibly dependent on a. No computer algebra system that we are aware of currently has sufficiently capable support for symbolic sets, leading us to develop a tool supporting symbolic sets using the Python symbolic computation library SymPy coupled with the Satisfiability Modulo Theories solver Z3.

The Ramsey problem R(3, k) seeks to determine the smallest value of n such that any red/blue edge coloring of the complete graph on n vertices must either contain a blue triangle (3-clique) or a red clique of size k. Despite its significance, many previous computational results for the Ramsey R(3, k) problem such as R(3, 8) and R(3, 9) lack formal verification. To address this issue, we use the software MathCheck to generate certificates for Ramsey problems R(3, 8) and R(3, 9) (and symmetrically R(8, 3) and R(9, 3)) by integrating a Boolean satisfiability (SAT) solver with a computer algebra system (CAS). Our SAT+CAS approach significantly outperforms traditional SAT-only methods, demonstrating an improvement of several orders of magnitude in runtime. For instance, our SAT+CAS approach solves R(3, 8) (resp., R(8, 3)) sequentially in 59 hours (resp., in 11 hours), while a SAT-only approach using state-of-the-art CaDiCaL solver times out after 7 days. Additionally, in order to be able to scale to harder Ramsey problems R(3, 9) and R(9, 3) we further optimized our SAT+CAS tool using a parallelized cube-and-conquer approach. Our results provide the first independently verifiable certificates for these Ramsey numbers, ensuring both correctness and completeness of the exhaustive search process of our SAT+CAS tool.

We enumerate all extended self-orthogonal diagonal Latin squares of order up to 10. Our method reduces the problem of enumerating extended self-orthogonal diagonal Latin squares to a satisfiability (SAT) problem, and we find all solutions of the SAT problem using a SAT solver. We additionally show that there is no triple of mutually orthogonal diagonal Latin squares of order 10 containing an extended self-orthogonal diagonal Latin square.

In this paper, we provide algorithmic methods for conducting exhaustive searches for periodic Golay pairs. Our methods enumerate several lengths beyond the currently known state-of-the-art available searches: we conducted exhaustive searches for periodic Golay pairs of all lengths v <= 72 using our methods, while only lengths v <= 34 had previously been exhaustively enumerated. Our methods are applicable to periodic complementary sequences in general. We utilize sequence compression, a method of sequence generation derived in 2013 by Dokovic and Kotsireas. We also introduce and implement a new method of "multi-level" compression, where sequences are uncompressed in several steps. This method allowed us to exhaustively search all lengths v <= 72 using less than 10 compute years. For cases of complementary sequences where uncompression is not possible, we introduce some new methods of sequence generation inspired by the isomorph-free exhaustive generation algorithm of orderly generation. Finally, we pose a conjecture regarding the structure of periodic Golay pairs and prove it holds in many lengths, including all lengths v < 100. We demonstrate the usefulness of our algorithms by providing the first ever examples of periodic Golay pairs of length v = 90. The smallest length for which the existence of periodic Golay pairs is undecided is now 106.

The queen domination problem asks for the minimum number of queens needed to attack all squares on an n× n chessboard. Once this optimal number is known, determining the number of distinct solutions up to isomorphism has also attracted considerable attention. Previous work has introduced specialized and highly optimized search procedures to address open instances of the problem. While efficient in terms of runtime, these approaches have not provided proofs that can be independently verified by third-party checkers. In contrast, this paper aims to combine efficiency with verifiability. We reduce the problem to a propositional satisfiability problem (SAT) using a straightforward encoding, and solve the resulting formulas with modern SAT solvers capable of generating proof certificates. By improving the SAT encoding with a novel literal ordering strategy, and leveraging established techniques such as static symmetry breaking and the Cube-and-Conquer paradigm, this paper achieves both performance and trustworthiness. Our approach discovers and corrects a discrepancy in previous results for n=16 and resolves the previously open case n=19.

One of the fundamental results in quantum foundations is the Kochen-Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known, the problem of finding the minimum KS vector system in three dimensions (3D) has remained stubbornly open for over 55 years. To address the minimum KS problem, we present a new verifiable proof-producing method based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS) that uses an isomorph-free orderly generation technique that is very effective in pruning away large parts of the search space. Our method shows that a KS system in 3D must contain at least 24 vectors. We show that our sequential and parallel Cube-and-Conquer (CnC) SAT+CAS methods are significantly faster than SAT-only, CAS-only, and a prior CAS-based method of Uijlen and Westerbaan. Further, while our parallel pipeline is somewhat slower than the parallel CnC version of the recently introduced Satisfiability Modulo Theories (SMS) method, this is in part due to the overhead of proof generation. Finally, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 40.3 TiB in order 23.

We show that the n'th digit of the base-b representation of the golden ratio is a finite-state function of the Zeckendorf representation of b^n, and hence can be computed by a finite automaton. Similar results can be proven for any quadratic irrational. We use a satisfiability (SAT) solver to prove, in some cases, that the automata we construct are minimal.

The Ramsey problem R(3,8) asks for the smallest n such that every red/blue coloring of the complete graph on n vertices must contain either a blue triangle or a red 8-clique. We provide the first certifiable proof that R(3,8) = 28, automatically generated by a combination of Boolean satisfiability (SAT) solver and a computer algebra system (CAS). This SAT+CAS combination is significantly faster than a SAT-only approach. While the R(3,8) problem was first computationally solved by McKay and Min in 1992, it was not a verifiable proof. The SAT+CAS method that we use for our proof is very general and can be applied to a wide variety of combinatorial problems.

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. This builds on work of van Frankenhuysen (1999) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt{2\delta/e}$ where $\delta$ is a constant such that all full-rank unimodular lattices of sufficiently large dimension $n$ contain a nonzero vector with $\ell_1$ norm at most $n/\delta$.

Cryptographic hash functions play a crucial role in ensuring data security, generating fixed-length hashes from variable-length inputs. The hash function SHA-256 is trusted for data security due to its resilience after over twenty years of intense scrutiny. One of its critical properties is collision resistance, meaning that it is infeasible to find two different inputs with the same hash. Currently, the best SHA-256 collision attacks use differential cryptanalysis to find collisions in simplified versions of SHA-256 that are reduced to have fewer steps, making it feasible to find collisions. In this paper, we use a satisfiability (SAT) solver as a tool to search for step-reduced SHA-256 collisions, and dynamically guide the solver with the aid of a computer algebra system (CAS) used to detect inconsistencies and deduce information that the solver would otherwise not detect on its own. Our hybrid SAT + CAS solver significantly outperformed a pure SAT approach, enabling us to find collisions in step-reduced SHA-256 with significantly more steps. Using SAT + CAS, we find a 38-step collision of SHA-256 with a modified initialization vector – something first found by a highly sophisticated search tool of Mendel, Nad, and Schläffer. Conversely, a pure SAT approach could find collisions for no more than 28 steps. However, our work only uses the SAT solver CaDiCaL and its programmatic interface IPASIR-UP.

The difficulty of factoring large integers into primes is the basis for cryptosystems such as RSA. Due to the widespread popularity of RSA, there have been many proposed attacks on the factorization problem such as side-channel attacks where some bits of the prime factors are available. When enough bits of the prime factors are known, two methods that are effective at solving the factorization problem are satisfiability (SAT) solvers and Coppersmith's method. The SAT approach reduces the factorization problem to a Boolean satisfiability problem, while Coppersmith's approach uses lattice basis reduction. Both methods have their advantages, but they also have their limitations: Coppersmith's method does not apply when the known bit positions are randomized, while SAT-based methods can take advantage of known bits in arbitrary locations, but have no knowledge of the algebraic structure exploited by Coppersmith's method. In this paper we describe a new hybrid SAT and computer algebra approach to efficiently solve random leaked-bit factorization problems. Specifically, Coppersmith's method is invoked by a SAT solver to determine whether a partial bit assignment can be extended to a complete assignment. Our hybrid implementation solves random leaked-bit factorization problems significantly faster than either a pure SAT or pure computer algebra approach.

This paper introduces AlphaMapleSAT, a novel Monte Carlo Tree Search (MCTS) based Cube-and-Conquer (CnC) SAT solving method aimed at efficiently solving challenging combinatorial problems. Despite the tremendous success of CnC solvers in solving a variety of hard combinatorial problems, the lookahead cubing techniques at the heart of CnC have not evolved much for many years. Part of the reason is the sheer difficulty of coming up with new cubing techniques that are both low-cost and effective in partitioning input formulas into sub-formulas, such that the overall runtime is minimized. Lookahead cubing techniques used by current state-of-the-art CnC solvers, such as March, keep their cubing costs low by constraining the search for the optimal splitting variables. By contrast, our key innovation is a deductively-driven MCTS-based lookahead cubing technique, that performs a deeper heuristic search to find effective cubes, while keeping the cubing cost low. We perform an extensive comparison of AlphaMapleSAT against the March CnC solver on challenging combinatorial problems such as the minimum Kochen-Specker and Ramsey problems. We also perform ablation studies to verify the efficacy of the MCTS heuristic search for the cubing problem. Results show up to 2.3x speedup in parallel (and up to 27x in sequential) elapsed real time.