10 problems

Let $v\colon\mathbb{N}\to\mathbb{Z}$ be the additive function defined by $v(p)=p$ for primes $p$ and $v(p^\alpha)=p+\alpha$ for prime powers $p^\alpha$ with $\alpha\geq2$. For inte…

Let $p>1$ be odd. A rational number $p/q$ has a palindromic quotient representation if there are palindromic numbers $A$ and $B$ such that $p/q=A/B$. Odd rational palindrome quotie…

Let $N$ be a natural number for which there exist palindromic numbers $A$ and $B$ with $N=A/B$, and measure the size of a solution by the length of the relevant base-$2$ representa…

For a natural number $N$, consider the heuristic algorithm used to search for palindromic numbers $A$ and $B$ satisfying $N=A/B$. Nontermination conjecture. There are infinitely ma…

Density-one conjecture. The asymptotic density of $T$ in $S$ is $1$. The paper proves that an explicitly described infinite family has $c(n)=\infty$, and asks for a simple way to d…

Simmons's conjecture. No integer $m$ exists such that $m^k$ is a palindromic number in the decimal base.

Simmons's conjecture. The fourth root of every palindromic fourth power is a palindromic number of the form $10^n+1$.

Let $\mathbb{P}$ be the set of palindromic natural numbers, and let $n$ be a sufficiently large natural number. The largest and second largest elements of $\mathbb{P}$ not exceedin…

Let $A$ be a positive integer and let $A^{*}$ denote its reversal. A pair $(A,A^{*})$ is polynomial if the multiplication $A \times A^{*}$ can be performed without carry, and…

Let $A$ and $B$ be positive integers. A pair $(A,B)$ is palindromic if the reversal of $A \times B$ equals the product of the reversals of $A$ and $B$, and it is polynomial if the…