7 problems

Ailon–Rudnick conjecture. If $a$ and $b$ are multiplicatively independent, then

Let $a>1$ and let $b$ be a non-zero integer. Hooley considered the sequence $2^n+5$ as $n$ runs over the positive integers. Hooley's conjecture. Almost all numbers in the sequence…

Limiting coprimality conjecture. The constant $c_{a,b}$ satisfies

Let $n$ be an integer. Erdős's conjecture. The digit $2$ occurs in the ternary expansion of $2^n$ for all $n>8$. This is a specific conjecture about the digit structure of powers o…

For the sequence $5\cdot2^n+1$, let $p_n$ denote its smallest prime divisor. The conjecture. The values $p_n$ are unbounded as $n$ varies. This is presented as a concrete instance…

Let $k$ be a power-free integer, and consider the sequence $k\cdot2^n+1$ as $n$ ranges over the natural numbers. Erdős's conjecture. Erdős's intuition is correct for all power-free…

For a positive integer $m$, let $P(m)$ denote its greatest prime factor. Erdős's conjecture. … This conjecture predicts that the greatest prime factor of $2^n-1$ grows faster than…