Related papers: On Polynomial Pairs of Integers

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…

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…

For a given base $g\ge2$, a positive integer is called a palindrome if its base $g$ expansion reads the same backwards as forwards. In this paper, we give an asymptotic formula for the number of relatively prime pairs of palindromes of a…

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…

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…

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite…

A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…

In this paper we prove that for any infinite word W whose set of factors is closed under reversal, the following conditions are equivalent: (I) all complete returns to palindromes are palindromes; (II) P(n) + P(n+1) = C(n+1) - C(n) + 2 for…

Given a set of integers, one can easily construct the set of their pairwise distances. We consider the inverse problem: given a set of pairwise distances, find the integer set which realizes the pairwise distance set. This problem arises in…

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2 = \Box$ (i.e., $a^2 + b^2$ is a square). A pythagorean pair $(a, b)$ is called a double-pythapotent pair if there is another pythagorean pair $(k,l)$ such that…

The paper proves two results involving a pair (A,B) of P-biisometric or (m,P)-biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair (A,B) is…

We introduce two new classes of integers. The first class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives…

For any integer $k$, M.Kaneko defined $k$-th poly-Bernoulli numbers as a kind of generalization of classical Bernoulli numbers using $k$-th polylogarithm. In case when $k$ is positive, $k$-th poly-Bernoulli numbers is a sequence of rational…

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

It is proven that the only integer solutions $(a,b)$ such that $a+b$ and $ab$ are palindromic are $(2,5\cdot 10^k-3)$, $(3,24)$ and $(9,9)$, and in a similar fashion, $b-a$ and $ab$ are only palindromic at $(a,b)=(3,147\cdot…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

A palintiple is a natural number which is an integer multiple of its digit reversal. A previous paper partitions all palintiples into three distinct classes according to patterns in the carries and then determines all palintiples belonging…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$,…

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…