Related papers: Perfect powers in Catalan and Narayana numbers

Let N_{a,b}(x) count the number of primes p<=x with p dividing a^k+b^k for some k>=1. It is known that asymptotically N_{a,b}(x) grows like c(a,b)x/log x for some rational number c(a,b) that depends in a rather intricate way on a and b. A…

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive…

The Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it…

Perfect error correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory.…

This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and…

A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$…

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by…

The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a…

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive…

A cubic (resp. biquadratic) theta series is a power series whose n-th coefficient is equal to 1 if n is a perfect cube (resp. fourth power) and zero otherwise. We improve on a result of Bradshaw by showing that such series is not a cubic…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

Nagata has conjectured that the following statement (N_r) holds for all $r\geq 10$: (N_r) if $P_1,...P_r \in {\mathbb P}^2$ are generic points then any plane curve $C$ satisfies $\sum_1^r mult_{P_i}(C)\leq \sqrt{r} deg(C)$. Nagata proved…

We prove that there exist infinitely many rationals a, b and c with the property that a^2-1, b^2-1, c^2-1, ab-1, ac-1 and bc-1 are all perfect squares. This provides a solution to a variant of the problem studied by Diophantus and Euler.

The existence of infinitely many consecutive prime triples $p_n$, $ p_{n+1}$, and $p_{n+2}$ as $n \to \infty$, is sufficient to prove that the Catalan constant $\beta(2)=0.9159655941\ldots $ is an irrational number. This note provides the…

A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms. These objects are enumerated by the well-known Catalan numbers, and have been introduced with the aim of giving a common language to most of the structures…

We introduce a variant of de Bruijn words that we call perfect necklaces. Fix a finite alphabet. Recall that a word is a finite sequence of symbols in the alphabet and a circular word, or necklace, is the equivalence class of a word under…

The problem of constructing a perfect cuboid is related to a certain class of univariate polynomials with three integer parameters $a$, $b$, and $u$. Their irreducibility over the ring of integers under certain restrictions for $a$, $b$,…

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems…

Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed…