Is it the case that for every non-negative integer $n$, iterating $n \to 2n+1$ eventually produces a prime number? (This is the same as asking whether for every positive integer $n$, there is a non-negative integer $k$ such that $2^k n – 1$ is prime.) If this is not settled by proof, is there some heuristic argument either way?
Does Iterating n to 2n+1 Always Produce a Prime Number?
number theoryprime numbers
Best Answer
The Riesel Numbers are odd numbers $k$ such that $k\cdot 2^N-1$ is never prime. It is known that there are infinitely many Riesel numbers. It follows that there are infinitely many $n$ such that your iteration, starting at $n$, produces no primes.
The smallest known Riesel number is $509203$, but there may be smaller ones. The Riesel numbers are the obscure cousins of the Sierpinski Numbers.