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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

 2021-05-14, 16:29 #12 Stargate38     "Daniel Jackson" May 2011 14285714285714285714 3·223 Posts 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
2021-05-14, 17:54   #13
kriesel

"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

3×5×373 Posts

Quote:
 Originally Posted by Stargate38 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

2021-05-14, 18:03   #14
charybdis

Apr 2020

33·17 Posts

Quote:
 Originally Posted by kriesel 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.

 2021-05-14, 21:02 #15 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 22·521 Posts 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)
2021-05-14, 21:43   #16
Viliam Furik

"Viliam Furík"
Jul 2018
Martin, Slovakia

28C16 Posts

Quote:
 Originally Posted by a1call 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.

2021-05-14, 23:07   #17
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

141628 Posts

Quote:
 Originally Posted by a1call 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.

2021-05-14, 23:50   #18
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

1,489 Posts

Quote:
 Originally Posted by a1call 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.

2021-05-15, 00:16   #19
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

952510 Posts

Quote:
 Originally Posted by R. Gerbicz 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

2021-05-15, 01:57   #20
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

1000001001002 Posts

Quote:
 Originally Posted by Viliam Furik 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

2021-05-15, 06:20   #21
LaurV
Romulan Interpreter

Jun 2011
Thailand

23·23·53 Posts

Quote:
 Originally Posted by chalsall There's no point in being pessimistic. It doesn't work, anyway...
Haha, good one! You made my day...

2021-05-28, 17:46   #22
tuckerkao

"Tucker Kao"
Jan 2020

Quote:
 Originally Posted by Stargate38 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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