Related papers: On a variant of Pillai's problem

For an integer $d\geq 2$ which is not a square, we show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2-dY^2=\pm 1$ which is a Tribonacci number, with a few exceptions that we completely…

We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this…

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative…

In the present note we prove a multiplicity result for a Kirchhoff type problem involving a critical term, giving a partial positive answer to a problem raised by Ricceri.

Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$…

The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…

The resolvability of equations in integers containing truncated Newton's binomial, is determined by the divisibility of the binomial by the characteristic parameters of the equation, which most often is the binomial exponent. Two types of…

In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent…

Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of…

In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation \begin{equation*} p^n - q^m = f. \end{equation*} We prove that for any non-constant polynomial $ f $ there are only finitely many…

We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…

Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular…

We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.

Let $k\ge 2$ and $\{F_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$--generalized Fibonacci numbers whose first $k$ terms are $0,\ldots,0,0,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all…

Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c,…

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the…

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…

The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a…

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.