Related papers: Iteration Steps of 3x+1 Problem

We present several new results involving $\Delta(x+U)-\Delta(x)$, where $U = o(x)$ and $$ \Delta(x):=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.

We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq…

In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$.…

This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ with positive initial conditions, being $a$ also a positive parameter. It is known that for $a=1$ all the sequences generated by this…

We consider the demixing problem of two (or more) high-dimensional vectors from nonlinear observations when the number of such observations is far less than the ambient dimension of the underlying vectors. Specifically, we demonstrate an…

Given a set $X$ of $n$ binary words of equal length $w$, the 3XOR problem asks for three elements $a, b, c \in X$ such that $a \oplus b=c$, where $ \oplus$ denotes the bitwise XOR operation. The problem can be easily solved on a word RAM…

We consider a linear recursion of the form $$R^{(k+1)}\stackrel{\mathcal D}{=}\sum_{i=1}^{N}C_iR^{(k)}_i+Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a real-valued random vector with $N\in\mathbb{N}=\{0, 1, 2, \dots\}$,…

We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi_{n+1}$. Here $M$ denotes a primitive matrix having…

We study random multidimensional assignment problems where the costs decompose into the sum of independent random variables. In particular, in three dimensions, we assume that the costs $W_{i,j,k}$ satisfy…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

Let $N_a$ be the number of solutions to the equation $x^{2^k+1}+x+a=0$ in $\GF {n}$ where $\gcd(k,n)=1$. In 2004, by Bluher \cite{BLUHER2004} it was known that possible values of $N_a$ are only 0, 1 and 3. In 2008, Helleseth and Kholosha…

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of…

An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and sufficient conditions on solutions to the…

While solving a special case of a question of Erd\H{o}s and Graham Steinerberger asks for all integers $n$ with $\phi(n)=\frac{2}{3} \cdot (n+1)$. He discovered the solutions $n\in\{5, 5 \cdot 7, 5\cdot 7\cdot 37, 5\cdot 7\cdot 37\cdot…

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n\ge 3k+1 can be…

This paper studies the problem of finding an $(1+\epsilon)$-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all…

In 1851 Prouhet showed that when $N=j^{k+1}$ where $j$ and $k$ are positive integers, $j \geq 2$, the first $N$ consecutive positive integers can be separated into $j$ sets, each set containing $j^k$ integers, such that the sum of the…

The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length $2(n+1)$ with exactly $n+1$ many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the…

Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq…

We study the problem of chasing positive bodies in $\ell_1$: given a sequence of bodies $K_{t}=\{x^{t}\in\mathbb{R}_{+}^{n}\mid C^{t}x^{t}\geq 1,P^{t}x^{t}\leq 1\}$ revealed online, where $C^{t}$ and $P^{t}$ are nonnegative matrices, the…