Related papers: Counting relatively prime pairs of palindromes

Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer…

We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free…

The reversal of a positive integer $A$ is the number obtained by reading $A$ backwards in its decimal representation. A pair $(A,B)$ of positive integers is said to be palindromic if the reversal of the product $A \times B$ is equal to the…

A palindrome in base $g$ is an integer $N$ that remains the same when its digit expansion in base $g$ is reversed. Let $g$ and $h$ be given distinct integers $>1$. In this paper we discuss how many integers are palindromes in base $g$ and…

A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every…

We study the distribution of palindromic numbers (with respect to a fixed base $g\ge 2$) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes $n\le x$ as $x\to\infty$. Our results show…

An integer $n\geq 1$ is a $v$-palindrome if it is not a multiple of $10$, nor a decimal palindrome, and such that the sum of the prime factors and corresponding exponents larger than $1$ in the prime factorization of $n$ is equal to that of…

In the year 2007, the author discovered an intriguing property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, prime factorize each number, and sum the numbers appearing in each…

Around the year 2007, one of the authors, Tsai, accidentally discovered a property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, which have prime factorizations $198 = 2\cdot…

It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large…

In this paper, we prove an asymptotic formula for the number of relatively prime pairs in the Piatetski-Shapiro sequence of arbitrarily large order. This improves the result of Pimsert, Srichan and Tangsupphathawat (2023), the order of…

For $g \ge 2$, we show that the number of positive integers at most $X$ which can be written as sum of two base $g$ palindromes is at most $\frac{X}{\log^c X}$. This answers a question of Baxter, Cilleruelo and Luca.

A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of…

We show that there exist exactly 203 positive integers $N$ such that for some integer $d \geq 2$ this number is a $d$-digit palindrome base 10 as well as a $d$-digit palindrome for some base $b$ different from 10. To be more precise, such…

For integer $g\ge 5$, we prove that any positive integer can be written as a sum of three palindromes in base $g$.

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…

A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a…

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

We prove new results related to the digital reverse $\overleftarrow{n}$ of a positive integer $n$ in a fixed base $b$. First we show that for $b\geq 26000$, there exists infinitely many primes $p$ such that $\overleftarrow{p}$ is…