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1. An algorithm and computation to verify Legendre’s conjecture up to $$7\cdot 10^{13}$$

   https://doi.org/10.1007/s40993-024-00589-4  

   Sorenson, Jonathan; Webster, Jonathan (March 2025, Research in Number Theory)

   Abstract We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integern, there exists a prime between$$n^2$$ $n^2$and$$(n+1)^2$$ ${(n+1)}^2$. Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between$$n^2$$ $n^2$and$$n(n+1)$$ $n(n+1)$and also between$$n(n+1)$$ $n(n+1)$and$$(n+1)^2$$ ${(n+1)}^2$. Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all$$n\le N$$ $n\le N$in time$$O( N \log N \log \log N)$$ $O(NlogNloglogN)$and space$$N^{O(1/\log \log N)}$$ $N^{O(1/loglogN)}$. We implemented a parallel version of our algorithm and improved the empirical verification of Oppermann’s conjecture from the previous$$N = 2\cdot 10^{9}$$ $N=2·{10}^9$up to$$N = 7.05\cdot 10^{13} > 2^{46}$$ $N=7.05·{10}^{13}>2^{46}$, so we were finding 27 digit primes. The computation ran for about half a year on each of two platforms: four Intel Xeon Phi 7210 processors using a total of 256 cores, and a 192-core cluster of Intel Xeon E5-2630 2.3GHz processors. 

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2. Linking Numbers in Three-Manifolds

   https://doi.org/10.1007/s00454-021-00287-3  

   Cahn, Patricia; Kjuchukova, Alexandra (July 2021, Discrete & Computational Geometry)

   Abstract LetMbe a connected, closed, oriented three-manifold andK, Ltwo rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number betweenKandLin terms of a presentation of Mas an irregular dihedral three-fold cover of $$S^3$$ $S^3$branched along a knot$$\alpha \subset S^3$$ $\alpha \subset S^3$. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $$\alpha $$ $\alpha$can be derived from dihedral covers of $$\alpha $$ $\alpha$. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. 

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3. On an Equivalence of Divisors on $$\overline {\text {M}}_{0,n}$$ from Gromov-Witten Theory and Conformal Blocks

   https://doi.org/10.1007/s00031-022-09752-6  

   Chen, L.; Gibney, A.; Heller, L.; Kalashnikov, E.; Larson, H.; Xu, W. (August 2022, Transformation Groups)

   Abstract We consider a conjecture that identifies two types of base point free divisors on$$\overline {\text {M}}_{0,n}$$ ${\overline{M}}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on$$\overline {\text {M}}_{0,n}$$ ${\overline{M}}_{0,n}$to the same statement forn= 4. A reinterpretation leads to a proof of the conjecture on$$\overline {\text {M}}_{0,n}$$ ${\overline{M}}_{0,n}$for a large class, and we give sufficient conditions for the non-vanishing of these divisors. 

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4. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (March 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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5. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (October 2023, Combinatorica)

   Abstract A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$ $P_5$-free graph with clique number$$\omega \ge 3$$ $\omega \ge 3$has chromatic number at most$$\omega ^{\log _2(\omega )}$$ ${\omega }^{{log}_2\left(\omega \right)}$. The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$ $(5/27)3^{\omega }$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$ $P_5$, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$ $P_5$-free graphs, and our result is an attempt to approach that. 

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