$2^i - 2293$ is always composite?

Question

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}}}

$$|2^n-2293|$$ is composite for $1\le n\le 200,000$, so a prime of the desired form must have more than $60,000$ digits. — Peter, Nov 18 '15 at 19:10
I conjectured that any odd number 2n+1=2^i+p, p is a (maybe negative) prime. however puzzled by 2293. It seems this conjecture is FALSE. http://2293.ml is my math site,(little info yet now!) — a boy, Nov 19 '15 at 17:06
I still tend to believe that there are primes (in fact infinite many primes) of the form $2^n-2293$. The situation is similar with Wieferich-primes, for example. They are unbelievable rare, but is believed that infinite many exist. But $2293$ is very tough indeed. :) — Peter, Nov 19 '15 at 18:54
@DanaJ , here is another project for you. Find a prime of the form $2^n-2293$ ! As you can see, I searched upto about $n=200,000$ without success. — Peter, Nov 19 '15 at 18:58
It appears that $2293$ is interesting for another, related reason: https://primes.utm.edu/curios/page.php/2293.html says: "The smallest number $k$ for which there is no known prime of the form $k\cdot2^n−1$." Note also that if one slightly changes the requirement, changing $2^i-2293$ to $2^i+2293$ then it becomes easy: $2^2+2293=2297$, a prime :) @user37238 — Mirko, Nov 20 '15 at 14:08
I am currently at $n=270,000$, no primes yet! A prime of the desired form must have more then $80,000$ digits! — Peter, Nov 20 '15 at 19:13
I have passed $n=300,000$ now. No primes found yet! I will continue upto $100,000$ digits. — Peter, Nov 21 '15 at 16:46
@Peter, I think that no primes of form |2^i-2293|. I see you are searching Least prime of the form 38^n+31 (http://adf.ly/1S7PWf). I also tend to believe no prime of the form 38^n+31 — a boy, Nov 22 '15 at 03:07
sagemath:300000%24 == 0 i = 300000 + 1 while not is_prime(2^i - 2293): print(i) i += 24 — a boy, Nov 22 '15 at 03:38
@a boy, please replace your commercialized link (at adf.ly) by the direct link http://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31 . After I have followed your link I have now a "cookie" from that people on my computer which I have to find and delete manually — Gottfried Helms, Nov 22 '15 at 07:37
@a boy as I already mentioned, I think that primes of both forms exists, unless there is a special reason that it is impossible. Of course, the smallest such prime can have millions of digits. But the functions grow slowly enough that heuristics lead to the conjecture that there are primes. — Peter, Nov 22 '15 at 10:35
I clicked the link above. Is this safe, or can I receive unwelcome cookies this way ? — Peter, Nov 22 '15 at 10:36
@gottfried helms how can I find out whether such cookies (only deletable manually) are on my computer ? — Peter, Nov 22 '15 at 10:38
@Peter: I just found that cookie using "Extras/Einstellungen/..." and then the register card for manual cookie-deletion.(I don't know for what reason/goal, but firefox uses one of the most uncomfortable windows I've ever seen for long lists to check/manipulate, so don't get exhausted too early) — Gottfried Helms, Nov 22 '15 at 10:43
i<=300121, no prime of the form 2^i-2293. It taken a whole afternoon to work out (300 000, 300 121) on cloud.sagemath.com. 3x @Peter — a boy, Nov 22 '15 at 13:09
@Gottfried Helms, I always neglect cookie. Before html5 born, there were cookies all over.the world. thanks for your kind attention — a boy, Nov 22 '15 at 13:13
According to my calculation, a prime of the desired form must have more than $100,000$ digits. Someone might check and approve this ... — Peter, Nov 24 '15 at 08:51
If there is a witness (a fixed product of primes) that is not coprime to $2^N - 2293$ for each natural $N$, then the period of $2$ in that witness is a multiple of either $3368903$ or $83550916$, based on the factors of $2^{49}-2293$ — will, Nov 25 '15 at 11:17
Turns out people have been looking for quite a while. E.g. Mersenneforum 2009 and Mersenneforum 2010. I haven't seen any results however. I've checked to n=360k, which is ~108k digits. — DanaJ, Mar 06 '16 at 00:16
This post from 2010: mersenneforum as well as an earlier one says it's been tested to over 700k. So over 210k digits. — DanaJ, Mar 06 '16 at 03:49
$2293$ is also the smallest unsolved candidate for a Riesel Number. — Vepir, Aug 16 '20 at 13:14

score 1 · Accepted Answer · answered Aug 16 '20 at 13:42

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.)

$2^i - 2293$ is always composite?

1 Answers1