Dozenal near- and quasi- repunit primes

[sweety439]

at n=12065

[tuckerkao]

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

[LaurV]

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

[Dr Sardonicus]

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...

[sweety439]

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.

[sweety439]

status for dozenal near-repdigit primes

[tuckerkao]

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.

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.

[sweety439]

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.

[LaurV]

Originally Posted by Dr Sardonicus

All but the first...

Yet, I did better than him!

[tuckerkao]

Originally Posted by sweety439

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.

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.