We show that, for all nonnegative integers k, Г, m and n, theYamabe invariant of #k(RP3)#Г(RP2 】 S1)#m(S2 】 S1)#n(S2. 】S1) is equal to the Yamabe invariant of RP3, provided k + Г ∶ 1. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 】 S1)#n(S2. 】S1),where S2. 】S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yam- abe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

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We prove that uniform random quadrangulations of the sphere with$n$faces, endowed with the usual graph distance and renormalized by$n$^{−1/4}, converge as$n$→$∞$in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called$Brownian map$, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of$geodesic stars$in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

For any natural number$m$(>1) let$P$($m$) denote the greatest prime divisor of$m$. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $$\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$$ (see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the$p$-adic theory of logarithmic forms.

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