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View Poll Results: Is M2133277 a prime?
Yes 3 11.11%
No 18 66.67%
Dunno 2 7.41%
Only in some bases 4 14.81%
Voters: 27. You may not vote on this poll

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Old 2021-05-14, 16:29   #12
Stargate38
 
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"Daniel Jackson"
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If interpreted as either Base-25 or Base-30, it's prime:

M2133277 (Base-25) = 134776610807 (Base-10)
M2133277 (Base-30) = 482624813017 (Base-10)

Obviously this goes on to infinity, so I won't list anymore. However, 2^2133277-1 isn't prime: https://www.mersenne.ca/exponent/2133277
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Old 2021-05-14, 17:54   #13
kriesel
 
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Quote:
Originally Posted by Stargate38 View Post
If interpreted as either Base-25 or Base-30, it's prime:

M2133277 (Base-25) = 134776610807 (Base-10)
M2133277 (Base-30) = 482624813017 (Base-10)
Hmm. I get something quite different.
2133277 (Base 25) = 499,267,057 (base ten), which is prime per https://www.alpertron.com.ar/ECM.HTM;
2133277 (Base 30) = 1,484,813,017 (base ten) which is divisible by 61, so the corresponding Mersenne number is composite also, divisible by M61.
(base conversions performed with https://calculator.name/base-conversion.php; confirmed with https://www.rapidtables.com/convert/...converter.html)
The corresponding Mersenne numbers would have ~150,000,000 and ~447,000,000 decimal digits, respectively, not 12.
M499267057 has no factor below 273 greater than 1.

Last fiddled with by kriesel on 2021-05-14 at 17:55
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Old 2021-05-14, 18:03   #14
charybdis
 
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Quote:
Originally Posted by kriesel View Post
Hmm. I get something quite different.
2133277 (Base 25) = 499,267,057 (base ten), which is prime per https://www.alpertron.com.ar/ECM.HTM;
2133277 (Base 30) = 1,484,813,017 (base ten) which is divisible by 61, so the corresponding Mersenne number is composite also, divisible by M61.
Looks like he took "M" to be a digit with value 22 (as one does), rather than a prefix indicating a Mersenne number.
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Old 2021-05-14, 21:02   #15
a1call
 
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Aren't all Mersenne-numbers with Prime-exponents, Fermat's-Probable-Prime in base 2^n?

Pari-GP:
Code:
forprime(n=1,19^1,{
    Mn=2^n-1;
    print(Mod(2,Mn)^(Mn-1););
    print(Mod(2^2,Mn)^(Mn-1););
    print(Mod(2^19,Mn)^(Mn-1););
})
Code:
Mod(1, 3)
Mod(1, 3)
Mod(1, 3)
Mod(1, 7)
Mod(1, 7)
Mod(1, 7)
Mod(1, 31)
Mod(1, 31)
Mod(1, 31)
Mod(1, 127)
Mod(1, 127)
Mod(1, 127)
Mod(1, 2047)
Mod(1, 2047)
Mod(1, 2047)
Mod(1, 8191)
Mod(1, 8191)
Mod(1, 8191)
Mod(1, 131071)
Mod(1, 131071)
Mod(1, 131071)
Mod(1, 524287)
Mod(1, 524287)
Mod(1, 524287)
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Old 2021-05-14, 21:43   #16
Viliam Furik
 
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Quote:
Originally Posted by a1call View Post
Aren't all Mersenne-numbers with Prime-exponents, Fermat's-Probable-Prime in base 2^n?
They are, but I don't see how that relates to the rest of the thread. However, I am not entitled to judge.
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Old 2021-05-14, 23:07   #17
retina
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Quote:
Originally Posted by a1call View Post
Aren't all Mersenne-numbers with Prime-exponents, Fermat's-Probable-Prime in base 2^n?
I guess that some of them are still probable primes, some are primes, but most are only pseudoprimes.
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Old 2021-05-14, 23:50   #18
R. Gerbicz
 
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"Robert Gerbicz"
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Quote:
Originally Posted by a1call View Post
Aren't all Mersenne-numbers with Prime-exponents, Fermat's-Probable-Prime in base 2^n?
That is true.
For any N if you consider the b bases for that N is a Fermat pseudoprime then these bases form a group in Z_N.
For Mersenne numbers this means that 2^n is such a base, because mp is a Fermat pseudoprime for base=2, fortunately
these means only p such bases, because 2^p==2^0 mod mp.

In an elementary way without group:
you need: (2^n)^(2^p-1)==2^n mod (2^p-1)

but we have: 2^p=a*p+2

hence: (2^n)^(2^p-1)==2^(n*(a*p+1))==2^n mod (2^p-1)
what we needed.

ps. this is the reason why we are using base=3 for Fermat testing the Mersenne numbers, base=2,4,8 etc is "bad". But you shouldn't fix base=3 to all numbers, because for other type of numbers: N could be a (trivial) pseudoprime for base=3, and you need to choose another base.
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Old 2021-05-15, 00:16   #19
Batalov
 
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Quote:
Originally Posted by R. Gerbicz View Post
But you shouldn't fix base=3 to all numbers, because for other type of numbers: N could be a (trivial) pseudoprime for base=3, and you need to choose another base.
Exactly right. Eisenstein-Mersennes, for example, or also some Zhou's 3-3-1 primes
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Old 2021-05-15, 01:57   #20
a1call
 
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Quote:
Originally Posted by Viliam Furik View Post
They are, but I don't see how that relates to the rest of the thread. However, I am not entitled to judge.
Well, posts number 18 & 19, and the current result of the pole are the reasons why Science is not a democratic process. I think everything relates.

Last fiddled with by a1call on 2021-05-15 at 01:59
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Old 2021-05-15, 06:20   #21
LaurV
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Quote:
Originally Posted by chalsall View Post
There's no point in being pessimistic. It doesn't work, anyway...
Haha, good one! You made my day...
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Old 2021-05-28, 17:46   #22
tuckerkao
 
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Quote:
Originally Posted by Stargate38 View Post
If interpreted as either Base-25 or Base-30, it's prime:

M2133277 (Base-25) = 134776610807 (Base-10)
M2133277 (Base-30) = 482624813017 (Base-10)

Obviously this goes on to infinity, so I won't list anymore. However, 2^2133277-1 isn't prime: https://www.mersenne.ca/exponent/2133277
There are several Mersenne Primes end in 77 when written in the dozenal base.

[dozenal]
MӾ5,077
M507,Ӿ77
M7,046,577
[/dozenal]

All of them got a 0 and a 5 somewhere in the number.

Last fiddled with by tuckerkao on 2021-05-28 at 17:47
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