discussion: For a squarefree composite n, the equations p | n/p-1 for every p | n are the local Giuga conditions, while p-1 | n-1 for every p | n are Korselt's local conditions for being Carmichael. A composite counterexample to Giuga's primality criterion would satisfy both systems at every prime factor. The search is restricted to odd primes because any composite Carmichael number is odd. Rather than demanding a presently unknown counterexample, this variant asks for the closest joint local fit in a fixed finite prime universe. Once a subset is supplied, all scores are verified by small modular reductions; the difficult step is proving that none of the roughly fourteen million competing 9-subsets has a better joint score. The lexicographic objective first balances the two systems, then rewards conditions holding at the same prime, so separate partial successes cannot masquerade as a near-counterexample.

domain: Computational number theory

inspiration: C114 — Giuga's primality conjecture

license: MIT

problem: Let P_30 be the set of the first 30 odd primes. Choose a 9-element subset S of P_30 and put n=product_{p in S} p. For p in S define G_p=1 when n/p is congruent to 1 modulo p, and C_p=1 when n is congruent to 1 modulo p-1. Put G(n)=sum_p G_p, C(n)=sum_p C_p, and B(n)=sum_p G_p C_p. Find S maximizing the lexicographic score (min(G(n),C(n)), B(n), G(n)+C(n)); subject to that, minimize n. Output S, n, the complete table of residues (n/p-1 mod p) and (n-1 mod p-1), the three scores, and the sets of primes satisfying each local condition. Supply an independently checkable branch-and-bound or pseudo-Boolean certificate proving optimality over all 9-subsets. Determine the two violation counts 9-G(n) and 9-C(n), the number of primes satisfying both local conditions, and the largest prime factor of n.

title: Closest bounded Giuga–Carmichael local fit


discussion: This searches for the most subdivision-poor positive instance at the first open chromatic boundary, rather than directly demanding a counterexample. If the optimum were zero, the winner would be a genuine 5-chromatic graph with no K_5 subdivision. Otherwise the exact minimum measures how deeply the required subdivision can be hidden. The admissible class is nonempty: two iterated Ore compositions of K_5 produce a thirteen-vertex 5-critical graph with clique number four. Verification is finite and transparent. Edge-criticality is certified by the deletion colorings, non-4-colorability by DRAT, and every topological K_5 can be enumerated as a subgraph with five degree-four branch vertices and degree-two internal vertices. The hard part is canonical enumeration of all admissible thirteen-vertex graphs and proof that no graph has fewer models.

domain: Structural and computational graph theory

inspiration: C128 — Hajós's conjecture for t=5 and t=6

problem: Let G range over simple graphs on thirteen vertices that are 5-critical, meaning chi(G)=5 while chi(G−e)=4 for every edge e, and require omega(G)=4. Let tau_5(G) be the number of edge subsets whose edge-subgraph is homeomorphic to K_5, with each subdivided K_5 counted once as an edge set. Find G minimizing tau_5(G). Subject to that, maximize girth, then minimize the number of edges, then minimize |Aut(G)|, and finally choose the canonical adjacency matrix. Output the edge list, a DRAT certificate that G is not 4-colorable, an explicit 4-coloring of G−e for every edge, and the complete list of topological K_5 subgraphs with their five branch vertices and ten branch paths. Supply an isomorph-free exhaustive certificate for the minimum. Determine tau_5(G), the orbit-size distribution of the subdivisions, girth, edge-connectivity, degree sequence, and automorphism-group order.

title: Subdivision-poor five-critical graph


discussion: This is a discrete Neumann hot-spots problem on lattice-convex planar domains. A positive value of h(P) means that both signs of the first nonconstant mode have all their hottest vertices strictly inside the discrete domain; h(P)=0 means that at least one extremum reaches the boundary. It is not a continuum counterexample, because the graph Laplacian is only a finite analogue of the Neumann Laplacian. Verification is exact: convexity and the lattice-point count are elementary, a Sturm sequence isolates lambda_2 and lambda_3, and signs of coordinates in Q(lambda_2) determine the extrema. The class is nonempty. For example, conv{(0,0),(8,0),(8,1),(3,5),(0,1)} has exactly 31 lattice points, a connected unit-grid graph, and an exact square-free nonzero Laplacian characteristic factor, hence simple lambda_2. The difficult step is proving the optimum over all normalized polygons, not checking a proposed winner.

domain: Spectral graph theory and discrete geometry

inspiration: C131 — Hot Spots conjecture for convex planar domains

problem: Let P range over two-dimensional convex lattice polygons contained in [0,8]^2, normalized so that min{x:(x,y) in P}=min{y:(x,y) in P}=0, and such that S(P)=P intersect Z^2 has exactly 31 points. Join two points of S(P) when their Euclidean distance is one, obtaining the unit-grid graph G(P), and require G(P) to be connected. Let L(P) be its combinatorial Laplacian and require the second eigenvalue lambda_2(P) to be simple. Put B(P)={v in S(P):deg_G(P)(v)<4}. If u is a lambda_2-eigenvector orthogonal to the constants, let E(P) be the union of the vertices where u is maximal and where it is minimal; because lambda_2 is simple, E(P) is independent of the sign chosen for u. Define h(P)=min_{v in E(P)} dist_G(P)(v,B(P)) and q(P)=sum_{v in E(P)} dist_G(P)(v,B(P)). Find an admissible P maximizing h(P). Subject to that, maximize q(P), then minimize |B(P)|, then minimize |Aut(G(P))|, and finally choose the canonical representative under lattice isometries. Output the polygon vertices, all 31 lattice points, the graph, the Laplacian characteristic polynomial, a minimal polynomial and rational isolating interval for lambda_2, an eigenvector with coordinates in Q(lambda_2), and exact sign-comparison certificates identifying E(P). Supply an isomorph-free exhaustive certificate for the global optimum. Determine h(P), q(P), the two extremal vertex sets, lambda_2, the spectral gap lambda_3-lambda_2, the graph diameter, the number of boundary vertices, and the automorphism-group order.

title: Interior hot spots on a convex lattice domain


discussion: After moving to the frame of the slowest of fourteen runners, the remaining thirteen relative speeds can be taken to be distinct positive integers, and the conjectural loneliness threshold is 1/14. This variant asks for the most dangerous primitive speed set below a fixed height, rather than for an arbitrary counterexample. If the optimum is below 1/14, the winning tuple is an actual counterexample; otherwise it is an exact finite near-miss benchmark. For fixed V, F_V is the lower envelope of finitely many triangular waves. Its maxima occur at rational breakpoints or intersections, and Delta(V)<=q is equivalent to a finite collection of near-integer intervals covering the unit circle. Hence a candidate and its exact value are easy to verify. The difficult component is excluding the enormous number of competing thirteen-subsets of {1,...,210}.

domain: Diophantine approximation and dynamics

inspiration: C134 — Lonely Runner conjecture, first unresolved case of 14 runners

problem: Let V={v_1<...<v_13} be a set of thirteen positive integers with v_13<=210 and gcd(v_1,...,v_13)=1. Define F_V(t)=min_{1<=i<=13} ||t v_i|| for t in R/Z, where ||.|| is distance to the nearest integer, and put Delta(V)=max_t F_V(t). Find V minimizing Delta(V). Subject to that, minimize v_13, then maximize the number of maximizing times in [0,1/2], then maximize the minimum number of active speeds at a maximizing time, and finally choose the lexicographically least tuple. Output V, Delta(V) as a reduced rational number, every maximizing time as a reduced fraction, and the set of active speeds at each such time. Give an exact interval-cover certificate proving F_V(t)<=Delta(V) for every t, together with a witnessing time proving equality, and supply a branch-and-bound or SAT certificate proving the global optimum over all primitive tuples in the stated box. Here the affine symmetry group means the group of real affine bijections x -> ax+b with a != 0 that preserve {0} union V setwise. Determine Delta(V), its comparison with 1/14, the number and denominator distribution of maximizing times, the active-set sizes, the largest gap between consecutive speeds, and the order of this affine symmetry group.

title: Most dangerous bounded fourteen-runner instance


discussion: For partitions of length at most five, the hive polytope has dimension at most six, so seven exact Littlewood–Richardson counts determine the entire stretching polynomial. A negative optimum would be a KTT counterexample inside a small box; a nonnegative optimum would identify the closest coefficient-level near-miss in that finite class. The primitive and degree conditions remove simple rescalings and linear examples. The class is nonempty: for example, lambda=(3,1), mu=(3,2,1), and nu=(4,3,2,1) have P(t)=(t+1)(t+2)/2. Candidate verification reduces to checking integer hives and rational interpolation, while global optimality requires enumerating and comparing all bounded partition triples.

domain: Algebraic combinatorics and representation theory

inspiration: C135 — King–Tollu–Toumazet positivity conjecture

problem: Let lambda, mu, and nu be partitions with at most five parts, largest part at most 12, |lambda|+|mu|=|nu|<=36, lambda lexicographically no larger than mu, and gcd of all their positive parts equal to one. Require the Littlewood–Richardson coefficient c_{lambda,mu}^{nu} to exceed one and the stretching polynomial P(t)=c_{t lambda,t mu}^{t nu} to have degree at least two. Write P(t)=a_0+a_1 t+...+a_d t^d over Q and put m(lambda,mu,nu)=min_{1<=i<=d} a_i. Find an admissible triple minimizing m. Subject to that, maximize the number of coefficients equal to m, then maximize d, then minimize |nu|, and finally choose the canonical triple after interchanging lambda and mu. Output the three partitions, the complete polynomial P, exact values P(0),...,P(6), the associated five-hive inequality system, and a rational generating-function or equivalent certificate for every lattice-point count used in the interpolation. Supply an exhaustive certificate over all bounded triples. Determine m, the complete coefficient vector and common denominator, the degree, the dimension and normalized volume of the hive polytope, its number of vertices, and the values P(1) and P(2).

title: Bounded King–Tollu–Toumazet stretching-positivity search