mersenneforum.org Dozenal near- and quasi- repunit primes
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2020-07-21, 14:49   #23
sweety439

Nov 2016

2,347 Posts

at n=12065
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2020-09-30, 00:59   #24
tuckerkao

Jan 2020

1708 Posts

Quote:
 Originally Posted by sweety439 Are there any searching for near- and quasi- repunit primes (primes of the form aaa...aaab, abbb...bbb, aaa...aaabc, abbb...bbbc, abccc...ccc, see thread https://mersenneforum.org/showthread.php?t=19717) in dozenal (duodecimal)? There are a lot of such searching in decimal (https://stdkmd.net/nrr/#factortables_nr and https://stdkmd.net/nrr/prime/primedifficulty.txt), and I finished this searching in dozenal up to n=1000 (decimal 1728)
5 out of the 6 largest known Mersenne Prime exponents are ended with 5 when written in the dozenal base. Also, I cannot find any 9s in all of them.

Ӿ,ӾƐ3,855
12,531,515
17,476,435
20,Ӿ28041
21,Ӿ46,Ɛ85
23,7ӾƐ,125

Last fiddled with by tuckerkao on 2020-09-30 at 01:06

 2020-09-30, 02:20 #25 LaurV Romulan Interpreter     Jun 2011 Thailand 2×5×883 Posts I can do better: when written in base 2, all mersenne prime's exponents end in 1.
2020-09-30, 02:28   #26
Dr Sardonicus

Feb 2017
Nowhere

2·52·71 Posts

Quote:
 Originally Posted by LaurV I can do better: when written in base 2, all mersenne prime's exponents end in 1.
All but the first...

2020-09-30, 03:12   #27
sweety439

Nov 2016

2,347 Posts

Quote:
 Originally Posted by tuckerkao 5 out of the 6 largest known Mersenne Prime exponents are ended with 5 when written in the dozenal base. Also, I cannot find any 9s in all of them. Ӿ,ӾƐ3,855 12,531,515 17,476,435 20,Ӿ28041 21,Ӿ46,Ɛ85 23,7ӾƐ,125
In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see Dozenal divisibility rule)

Also, these project is for the near-repunit and quasi-repunit primes in dozenal, not for the Mersenne Prime exponents in dozenal.

 2020-09-30, 03:13 #28 sweety439     Nov 2016 2,347 Posts
2020-09-30, 03:35   #29
tuckerkao

Jan 2020

23·3·5 Posts

Quote:
 Originally Posted by sweety439 In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see Dozenal divisibility rule) Also, these project is for the near-repunit and quasi-repunit primes in dozenal, not for the Mersenne Prime exponents in dozenal.
I was mentioning about no 9s for the entire numbers not only the ending units.

For example 9 dozen 1 and 9 dozen 5 are both primes.

Quote:
 Originally Posted by LaurV I can do better: when written in base 2, all mersenne prime's exponents end in 1.
The 0 enders = even numbers, the 1 enders = odd numbers which sound very familiar to everyone.

Base 4 will give more insights as whether the prime exponents turn out to be the 1 ender or the 3 ender.

Last fiddled with by tuckerkao on 2020-09-30 at 03:51

2020-09-30, 04:46   #30
sweety439

Nov 2016

2,347 Posts

Quote:
 Originally Posted by tuckerkao I was mentioning about no 9s for the entire numbers not only the ending units. For example 9 dozen 1 and 9 dozen 5 are both primes. The 0 enders = even numbers, the 1 enders = odd numbers which sound very familiar to everyone. Base 4 will give more insights as whether the prime exponents turn out to be the 1 ender or the 3 ender.
Well, there is a list for all Mersenne primes and all Mersenne exponents in dozenal: https://dozenal.fandom.com/wiki/Mersenne_prime

All Mersenne primes > 3 end with 7, and all Mersenne primes > 7 end with either 27 or X7 (27 and X7 are the only two-digit Mersenne primes).
Also, Mersenne exponents end with E are fewer than Mersenne exponents end with 1, 5, or 7, since if p end with E and 2p+1 is also prime (e.g. p = E, 1E, 6E, XE), then Mp is divisible by 2p+1, thus composite.

Last fiddled with by sweety439 on 2020-09-30 at 04:49

2020-09-30, 04:50   #31
LaurV
Romulan Interpreter

Jun 2011
Thailand

883010 Posts

Quote:
 Originally Posted by Dr Sardonicus All but the first...
Yet, I did better than him!

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