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Old 2020-02-08, 07:45   #1
tuckerkao
 
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Default Mersenne numbers in Dozenal base

It seems like when 2n - 1 expressions are written in Dozenal base, the repetitive patterns of the ending digits show up with a formula: see attached thumbnail.
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Old 2020-02-10, 14:12   #2
LaurV
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The order of 3 mod 2 is 2 and the order of 5 mod 2 is 4, the order of 9 is 6, etc, so there is no wonder that every second mersenne is multiple of 3, and every 4th is multiple of 5, etc. In your case, the base is multiple of 3 too, so what you see is just an elementary modular property.

Think about why we only test prime exponents, from the perspective of representing Mp in other bases than 10 (like, 2, 3, 6, 12, 15, etc). For example, if you represent them in base 5, or 15, what would happen with the last digit every 4 numbers?

Last fiddled with by LaurV on 2020-02-10 at 14:13
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Old 2020-02-10, 18:03   #3
tuckerkao
 
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Quote:
Originally Posted by LaurV View Post
Think about why we only test prime exponents, from the perspective of representing Mp in other bases than 10 (like, 2, 3, 6, 12, 15, etc). For example, if you represent them in base 5, or 15, what would happen with the last digit every 4 numbers?
In base 5, it'll repeat with 3, 1, 2, 4 and in decimal,it;'ll repeat on 2, 4, 8, 6.

I prefer to count in Dozenal since they repeat more often, so I have to memorize less digits after identifying the more frequent ending digit pattterns.

Any 2n * m - 1 will be the multiples of 2n - 1, so that's why n has to be a prime in order for the whole result to be a prime too.

Last fiddled with by tuckerkao on 2020-02-10 at 18:05
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Old 2020-02-11, 02:57   #4
sweety439
 
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In dozenal, except 3 and 7, all Mersenne primes ends with 27 or X7 (the only two 2-digit Mersenne primes), this is a list of first 27 (decimal 31) Mersenne primes in dozenal

Dozenal is my favorite base, it also has these properties:

* All squares end with square digits (0, 1, 4, 9).
* All primes end with prime digits (2, 3, 5, 7, E) or 1. (equivalently, all primes >=5 end with prime digits >=5 (5, 7, E) or 1).
* All numbers k such that the negative-Pell equation x^2-k*y^2=-1 is solvable end with all such single-digit number k (1, 2, 5, X).
* Except 0 = F0 and 1 = F1 = F2, the only square Fibonacci number is 100 = F10 (100 is exactly the square of 10), thus, 10 is the only base such that 100 is a Fibonacci number (since 100 in a base is just the square of this base, and 0 and 1 cannot be the base of numeral system)
* A Fibonacci number can end with any digit but 6, and if a Fibonacci number ends with 0, then it must end with 00.
* The period of the final digit of Fibonacci number is 20, that of the final two digits is also 20 (dozenal is the largest base such that the period of the final digit of Fibonacci number is the same as that of the final two digits of Fibonacci number, if there are no Wall-Sun-Sun primes). For n >= 2, the period of the final n digits of Fibonacci number is 2*10^(n-1). (2 followed by n-1 zeros, which is an n-digit number 200...000)

See https://dozenal.fandom.com/wiki/Properties_of_dozenal for more properties of the dozenal base, I am interested in it :))
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Old 2020-02-11, 03:02   #5
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Problem: In dozenal, 21 and 201 are squares, prove or disprove they are the only squares of the form 2000...0001 in dozenal.
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Old 2020-02-11, 03:06   #6
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Quote:
Originally Posted by sweety439 View Post
Problem: In dozenal, 21 and 201 are squares, prove or disprove they are the only squares of the form 2000...0001 in dozenal.
What has this got to do with Mersenne numbers? If you want to start your own thread on dozenal numbers, feel free to do so.
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Old 2020-02-11, 03:28   #7
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Quote:
Originally Posted by axn View Post
What has this got to do with Mersenne numbers? If you want to start your own thread on dozenal numbers, feel free to do so.
Well, my problem is whether 2*12^n+1 can be square for n other than 1 and 2

Last fiddled with by sweety439 on 2020-02-11 at 03:28
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Old 2020-02-11, 04:44   #8
tuckerkao
 
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Quote:
Originally Posted by sweety439 View Post
Well, my problem is whether 2*12^n+1 can be square for n other than 1 and 2
For your question, you can use the factorizer, see if any exponent from 3 up will return you a square root integer.
https://www.numberempire.com/numberfactorizer.php
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