Related papers: A solution of $3x+1$ problem
A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and $\sum_{i=1}^k…
Let $p$ and $q$ be distinct primes such that $q+1 | p-1$. In this paper we find all integer solutions $a$, $b$ to the equation $1/a + 1/b = (q+1)/pq$ using only elementary methods.
The first numerical solution of the 3-D travel time tomography problem is presented. The globally convergent convexification numerical method is applied.
In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables…
Consider the diophantine equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$ in positive integers $x_1\le x_2$, and $y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence, respectively. In this…
The group theoretical description of the three-particle problem provides successful techniques for the solution of different questions. We present here a review of this approach.
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 $3\pmod{4}$.…
We introduce and study a new kind of congruent number problem on the right trapezoid.
In this paper we give a closed expression for a multiple factorial series, solving Open Problem 3.137 in a recent book by O. Furdui. The method is based on properties of divided differences. It applies also to similar series and certain…
In this paper we provide an application to the Neumann problem of a recent three critical points theorem.
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd,…
In a recent work, the present author developed an efficient method to find the number of solutions of $ax+by+cz=n$ in non-negative integer triples $(x,y,z)$ where $a,b,c$ and $n$ are given natural numbers. In this note, we use that formula…
We provide infinitely many solutions of a Dirichlet problem on balls.
In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a…
We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.
We continue our recent work on averages for ternary additive problems with powers of prime numbers.
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…
We provide answers to natural combinatorial questions which arise from learning the rules to Tak. We solve the discarded cases of the $3\times 3$ game not previously considered by Joshua Achiam, showing that Player $1$ has a forced win from…
Problem 1325 from the journal Crux Mathematicorum is revisited, and a new solution is presented.
The aim of this paper is to construct a new expansion of $(1+1/x)^x$ related to Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001) 651-659].