at n=12065

[QUOTE=sweety439;545220]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 [URL="https://mersenneforum.org/showthread.php?t=19717"]https://mersenneforum.org/showthread.php?t=19717[/URL]) in dozenal (duodecimal)?

There are a lot of such searching in decimal ([URL="https://stdkmd.net/nrr/#factortables_nr"]https://stdkmd.net/nrr/#factortables_nr[/URL] and [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]https://stdkmd.net/nrr/prime/primedifficulty.txt[/URL]), and I finished this searching in dozenal up to n=1000 (decimal 1728)[/QUOTE]
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

I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.

[QUOTE=LaurV;558296]I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.[/QUOTE]
All but the first...

[QUOTE=tuckerkao;558291]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[/QUOTE]

In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see [URL="https://dozenal.fandom.com/wiki/Divisibility_rule"]Dozenal divisibility rule[/URL])

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

[URL="https://dozenal.fandom.com/wiki/Near-repdigit_prime"]status for dozenal near-repdigit primes[/URL]

[QUOTE=sweety439;558303]In dozenal, no primes end with 9, since all numbers end with 0, 3, 6, 9 are divisible by 3 (see [URL="https://dozenal.fandom.com/wiki/Divisibility_rule"]Dozenal divisibility rule[/URL])

Also, these project is for the [B]near-repunit and quasi-repunit[/B] primes in dozenal, not for the Mersenne Prime exponents in dozenal.[/QUOTE]
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=LaurV;558296]I can do better: when written in base 2, [U][B]all[/B][/U] mersenne prime's exponents end in 1.[/QUOTE]
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.

[QUOTE=tuckerkao;558310]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.[/QUOTE]

Well, there is a list for all Mersenne primes and all Mersenne exponents in dozenal: [URL="https://dozenal.fandom.com/wiki/Mersenne_prime"]https://dozenal.fandom.com/wiki/Mersenne_prime[/URL]

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.

[QUOTE=Dr Sardonicus;558298]All but the first...[/QUOTE]
Yet, I did better than him! :razz:

[QUOTE=sweety439;558318]Well, there is a list for all Mersenne primes and all Mersenne exponents in dozenal: [URL="https://dozenal.fandom.com/wiki/Mersenne_prime"]https://dozenal.fandom.com/wiki/Mersenne_prime[/URL]

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.[/QUOTE]
Thanks for the list, it seems like when the exponents end in dozenal 5, it has the slightly higher chance for being a Mersenne Prime.

I have my list for the exponents in dozenal enders, Red for 1, Blue for 5, Pink for 7, Skyblue for Ɛ.