Related papers: A note on the Erd\H{o}-Straus Conjecture

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive…

The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in…

A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked…

Let $E/\mathbb{Q}$ be an elliptic curve of level $N$ and rank equal to $1$. Let $p$ be a prime of ordinary reduction. We experimentally study conjecture $4$ of B. Mazur and J. Tate in his article "Refined Conjectures of the Birch and…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…

The paper substantiates the conjecture of the asymptotic behavior of the largest distance between consecutive primes: $sup_{p_i \leq x}(p_{i+1}-p_i) \sim 2e^{-\gamma} \log^2(x)$, where $\gamma$ is the Euler constant. The Hardy-Littlewood…

A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant $\Delta_E$. For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$ and write $N_p=p+1-a_p$. In 1965, Birch and Swinnerton-Dyer formulated a conjecture…

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f…

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$.…

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional…

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

We upgrade [1] to a complete proof of the conjecture NP = PSPACE. [1]: L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55-83 (2019)

Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime…

Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In…

In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

In this note we announce the proof of the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s => 3; this is new for s => 4, the cases s = 1,2,3 having been previously established. More precisely we outline a proof (details of which…

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_{\nu}+s)\equiv (y_1+s)...(y_{\nu}+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here,…