$2^i - 2293$ is always composite?

Is $2^i - 2293$ always composite for $i=1,2,3,...$ ?

I have known: if $2^i - 2293$ is prime, $i$ must have the form $i = 24 k+1$

In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}]

Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943, 1}}, {{6737807, 1}, {83550917, 1}}, {{399550573, 1}, {23638391743063, 1}}, {{281, 1}, {14821, 1}, {24203, 1}, {3712421, 1}, {423447263633, 1}}, {{149, 1}, {9492181, 1}, {1879650895890301462105483811, 1}}, {{137, 1}, {2683, 1}, {2360851, 1}, {2808601, 1}, {2020240309, 1}, {9058295304389951, 1}}, {{23, 1}, {29, 1}, {107, 1}, {199, 1}, {21035159, 1}, {5797034797, 1}, {28376991193, 1}, {15226094729816791, 1}}, {{526557780757, 1}, {1946642765756893, 1}, {12247765663995514289321022531499, 1}}, {{47, 1}, {617, 1}, {160191103, 1}, {8207681257, 1}, {9477520181923, 1}, {46405673331331, 1}, {12560339159195827, 1}}, {{439, 1}, {80494171516099513876232232380087403910135940632146649572738323\ 52130381, 1}}}

Solution 1:

This is a part of The dual Riesel problem, and is an open problem.

This problem is related to The Riesel problem, which consists in determining the smallest Riesel number: a number $k$ such that $k\cdot 2^n - 1$ is not prime for any integer $n$. The smallest unsolved candidate for this is $2293$. (The smallest known Riesel number is $509203$.)

The dual Riesel numbers are defined as the odd natural numbers $k$ such that $|2^n - k|$ is composite for all natural numbers $n$. The smallest unsolved candidates for these numbers are $1871$ and $2293$.

There is a conjecture that the set of dual Riesel numbers is the same as the set of Riesel numbers.

The problem of Riesel numbers dates back to year $1956$. (post on Primegrid.)