Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)
Abstract We show that for some even $k\leqslant 3570$ and all  $k$ with $442720643463713815200|k$, the equation $\phi (n)=\phi (n+k)$ has infinitely many solutions $n$, where $\phi$ is Euler’s totient function. We also show that for a positive proportion of all $k$, the equation $\sigma (n)=\sigma (n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.
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NSF-PAR ID:
10252047
Journal Name:
International Mathematics Research Notices
ISSN:
1073-7928
Positive$$k\mathrm{th}$$$k\mathrm{th}$-intermediate Ricci curvature on a Riemanniann-manifold, to be denoted by$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when$$k =1$$$k=1$and$$k=n-1$$$k=n-1$respectively). In this work, we produce many examples of manifolds of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$withksmall by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension$$n\ge 7$$$n\ge 7$congruent to$$3\,{{\,\mathrm{mod}\,}}4$$$3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$for some$$k$k. We also prove that each dimension$$n\ge 4$$$n\ge 4$congruent to 0 or$$1\,{{\,\mathrm{mod}\,}}4$$$1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports closed manifolds which carry metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$with$$k\le n/2$$$k\le n/2$, but do not admit metrics of positive sectional curvature. 2. Abstract The elliptic algebras in the title are connected graded \mathbb {C} -algebras, denoted Q_{n,k}(E,\tau ) , depending on a pair of relatively prime integers n>k\ge 1 , an elliptic curve E and a point \tau \in E . This paper examines a canonical homomorphism from Q_{n,k}(E,\tau ) to the twisted homogeneous coordinate ring B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k}) on the characteristic variety X_{n/k} for Q_{n,k}(E,\tau ) . When X_{n/k} is isomorphic to E^g or the symmetric power S^gE , we show that the homomorphism Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k}) is surjective, the relations for B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k}) are generated in degrees \le 3 and the noncommutative scheme \mathrm {Proj}_{nc}(Q_{n,k}(E,\tau )) has a closed subvariety that is isomorphic to E^g or S^gE , respectively. When X_{n/k}=E^g and \tau =0 , the results about B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k}) show that the morphism \Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1} embeds E^g as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces. 3. Abstract. This work measured  \mathrm{d}\sigma/\mathrm{d}\Omega d σ / d Ω for neutral kaon photoproduction reactions from threshold up to a c.m. energy of 1855MeV, focussing specifically on the  \gamma p\rightarrow K^0\Sigma^+ γ p → K 0 Σ + ,  \gamma n\rightarrow K^0\Lambda γ n → K 0 Λ , and  \gamma n\rightarrow K^0 \Sigma^0 γ n → K 0 Σ 0 reactions. Our results for  \gamma n\rightarrow K^0 \Sigma^0 γ n → K 0 Σ 0 are the first-ever measurements for that reaction. These data will provide insight into the properties of  N^{\ast} N * resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the  \pi N π N channel. Integrated cross sections were extracted by fitting the differential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions. 4. Abstract Given two k -graphs ( k -uniform hypergraphs) F and H , a perfect F -tiling (or F -factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H . For all complete k -partite k -graphs K , Mycroft proved a minimum codegree condition that guarantees a K -factor in an n -vertex k -graph, which is tight up to an error term o ( n ). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K (k) (1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively. 5. In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence {\mathcal{D}} , we obtain a sharp criterion such that for almost every \unicode[STIX]{x1D6FC} the inequality$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$has infinitely many coprime solutions (n,p)\in \mathbb{N}\times \mathbb{Z} for a certain one-parameter family of \unicode[STIX]{x1D713} . Also, under a minor condition on pseudo-absolute-value sequences {\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k} , we obtain a sharp criterion on a general sequence \unicode[STIX]{x1D713}(n) such that for almost every \unicode[STIX]{x1D6FC} the inequality$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray} has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ .