Mersenne Prime or not? (Is that a question?)

[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

[kriesel]

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.

[charybdis]

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.

[a1call]

Aren't all Mersenne-numbers with Prime-exponents, Fermat's-Probable-Prime in base 2^n?

Pari-GP:

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););
})

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)

[Viliam Furik]

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.

[retina]

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.

[R. Gerbicz]

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.

[Batalov]

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

[a1call]

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.

[LaurV]

Originally Posted by chalsall

There's no point in being pessimistic. It doesn't work, anyway...

Haha, good one! You made my day...

[tuckerkao]

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.