20220511, 23:50  #1 
May 2022
9_{10} Posts 
Pseudo Fact  Last Digits of Mersenne Primes
Hi,
See attached. 4423 and 43112609(marked in blue) are only exponents so far, where the last 39 digits lacked one of the digits. 4423 had no 4s 43112509 had no 3s 
20220518, 02:31  #2  
Feb 2017
Nowhere
2×17×173 Posts 
Quote:
I suspect there is, but don't have a clue how to approach the question. I did a bit of checking. For k = 100, lift(Mod(2,10^100)^1088) lacks the decimal digit 2, so k > 100. Lift(Mod(2,10^100)^n) lacks at least one decimal digit for 30 values of n, 334 < n < 100000. This does not fill me with optimism regarding a bruteforce assault on the question. I checked for k = 200 up to n = 10^7 without finding any 200digit endings lacking any of the decimal digits, but 10^7 is a tiny, tiny fraction of 4*5^199, the multiplicative order of 2 (mod 5^200). The largest power of 2 I know of whose complete decimal expansion lacks any of the decimal digits is 2^168 = 374144419156711147060143317175368453031918731001856 which lacks the decimal digit 2. Is it in fact the largest? I don't have a clue how to approach that question, either. 

20220519, 16:00  #3 
May 2022
11_{8} Posts 
Extra pseudofact: I tried to analyze those digits under many amateur criteria, and there is one sequence that doesn't show up in any of those 40 Mersenne last 39digit numbers. It only cuts work by 1/8th but still (Only need to check exponents that hold the sequence after last 39 digits are generated).
With this being said, can we treat last ndigits of Mersenne primes as random digits to see if there is any oddity even at small sample size of 40 #s? P.S. Obviously I'm not including 1st digit (last in fact) as it will be odd :) Last fiddled with by Graph2022 on 20220519 at 16:03 
20220519, 21:33  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
26AD_{16} Posts 

20220520, 18:06  #5 
May 2022
3^{2} Posts 
That only 1/8th of all exponents needs to be checked. This is mostly for people that like to hit that 100 Million or 1 Billion digit mark or even possibly 10 Billion (who knows when it hits LOL).
I assume that eventually it needs to happen. IF not, i'm curious why it doesn't happen (that sequence) that haven't happened for 40 times in a row(assuming that last 40 digits of Mersenne #s are ,,random"). Last fiddled with by Graph2022 on 20220520 at 18:09 
20220520, 23:46  #6 
May 2022
3^{2} Posts 
Extra pseudo fact. This time on bellwether.
M9689 and M25,964,951 have a 17 streak ( the longest so far) where digits (whether odd or even) correspond to each other (aka Bellwether). 
20220520, 23:50  #7 
If I May
"Chris Halsall"
Sep 2002
Barbados
3^{2}×5^{2}×47 Posts 

20220521, 00:00  #8 
May 2022
3^{2} Posts 
Lol, no i do not. Don't worry though, I already requested for thread to be removed so that you are happier :)
For some reason OP of the thread can't remove it after certain time, or i somehow can't see the option. 
20220521, 13:58  #9  
Feb 2017
Nowhere
2·17·173 Posts 
Quote:
As mentioned in this recent post, a Mersenne prime can terminate in at most 3 1's. 

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