7 problems

Let $b\geq 2$ be a fixed base. For a positive integer, write $\overleftarrow{n}$ for its digit reversal in base $b$. Then $p_1$ and $p_2$ are primes, and…

Hasler's structural infinitude conjecture. The equation $\binom{3n}{n} \equiv 3^n \pmod{n}$ has infinitely many solutions of the form $n=3^k\cdot p$, where $k$ is a positive intege…

For a fixed base $b\geq 2$, let $\overleftarrow{K}_b$ be the smallest number greater than $1$ such that every sufficiently large integer is a sum of at most $\overleftarrow{K}_b$ r…

For a fixed base $b\geq 2$, define the Nnamlerinchs constant $\overleftarrow{K}_b$ to be the smallest number greater than $1$ such that every sufficiently large integer can be writ…

Work in base $b=10$, and write $\overleftarrow{p}$ for the digit reversal of a prime $p$. Base-10 ternary reversed-prime conjecture. Every sufficiently large odd integer $N$ can be…

Let $b$ be a fixed prime, and let $\overleftarrow{p}$ denote the digit reversal of a prime $p$ in base $b$. Two-reversed-prime conjecture. Every sufficiently large even integer $N$…

Let $n$ be an integer with $n\ge 9$, and consider the base-$3$ expansion of $2^n$. Erdős's ternary conjecture. For every $n\ge 9$, this expansion contains the digit $2$. This class…