Related papers: Remark on a paper by Wu
We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by…
In this paper, using quaternion arithmetic in the ring of Lipschitz integers, we present a proof of Zh\`i-W\v{e}i S\={u}n's "1-3-5 conjecture" for integral solutions, and for all natural numbers greater than a specific constant. This,…
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
Recently, Miller and Wu introduced the positive $\lambda$-calculus, a call-by-value $\lambda$-calculus with sharing obtained by assigning proof terms to the positively polarized focused proofs for minimal intuitionistic logic. The positive…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].
We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs…
A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number…
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…
In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\|W\|_{L^{\infty}({\mathbf…
We give a sketch for an alternative proof of a recent result by J. Tseng.
We prove two positivity conjectures proposed by Guo for alternating sums and factorial ratios built from Gaussian coefficients. The first result proves the positivity of the odd $q$-super Catalan numbers \[…
Recently, Wang and Ma propose a conjecture associated with the possible generalization of Andrews-Warnaar identities. It is confirmed in this paper. As the applications of this conjecture, we prove that a family of series can be expressed…
We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of…
Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.
In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.
In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.
We introduce and study expansions of real numbers with respect to two integer bases.
Not only a review of Weintraub's Differential Forms: Theory and Practice but also a discussion of why differential forms should be taught to undergraduates and an overview of some of the other possible texts that could be used.
We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop…