Given an initial integer $x\_0>0$, one can consider the first prime of the recursive sequence $x\_i=1+2x\_{i-1}$.


Naïvely such a prime should exist for $x\_0$ arbitrary since the sequence $\log(x\_i)$ is asymptotically
an arithmetic progression. Sometimes it takes however some
time: for $x\_0=147$ my Maple algorithm stops when hitting a 771-digit number
labelled as prime by Maple (Maple does not use a primality proof but
some strong primality tests if I am not mistaken).


Starting with $x\_0=658$ I lost patience: No prime among the first
$56000$ iterates leading to numbers with almost $17000$ digits.


I tried to find an easy reason: If $x\longmapsto 2x+1$
is $k$-periodic for $k$ prime numbers $p\_0,\dotsc,p\_{k-1}$ such that $x\_i\equiv 0\pmod{p\_i}$ and $x\_i>p\_i$ then there is obviously no prime in this sequence. Such an easy argument fails for all small values of $k$ for the sequence $x\_0=658,\dotsc$.


Perhaps my patience ran out a bit early and the sequence will hit
eventually a prime:


*Does an integral sequence given by $x\_0>0$, $x\_i=1+2x\_{i-1}$ necessarily contain a prime number?*



First prime of the form $x_i$ for $x_0=658$ and $x_i=1+2x_{i-1}$
The $n$th term of your sequence is $x\_{n} = 659 \cdot 2^{n} - 1$. People have long searched for prime values of numbers $k \cdot 2^{n} - 1$ for small $k$, with the goal of proving that $k = 509203$ is the *smallest* positive integer for which $k \cdot 2^{n} - 1$ is composite for all $n$. (This is called the Riesel problem.)


According to the MathWorld entry on [Riesel numbers](https://mathworld.wolfram.com/RieselNumber.html), in 2004 Dave Linton discovered that $659 \cdot 2^{800516} - 1$ is prime, and that $800516$ is the smallest $n$ so that $659 \cdot 2^{n} - 1$ is prime. A lot of patience would have been required indeed.


Currently, the smallest $k$ for which it is unknown if $k \cdot 2^{n} - 1$ is ever prime is $k = 23669$ and a distributed computing project related to this is run by [PrimeGrid](https://www.primegrid.com/stats_trp_llr.php). For $k = 23669$ all $n < 13968972$ have been tested.


