2^primenumber -1 is always a prime number ?
 

Hi All,

Please let me know was if 2^8191 -1 proved to be not prime number ?
Since I saw the symmetry till the number ( 2^127-1 ) that 2 power of any primenumber is a primenumber too.


Regards,
Mastan

2^8191 -1 has factors 338193759479, 210206826754181103207028761697008013415622289, so it cannot be prime. End of proof.

Ok, Thanks what about this .


2 ^ 2 ^127 -1 which is more clearly POWER(2,(POWER(2,127)) -1

[QUOTE=Mastan;327845][B]2^primenumber -1 is always a prime number ?[/B][/QUOTE]

No, sorry. All the Mersenne numbers we are checking for primality have this form and almost all are proved to be composite in the end. So far we have only found 48 numbers of this form who are prime.

Check:

[URL]http://en.wikipedia.org/wiki/Mersenne_prime#List_of_known_Mersenne_primes[/URL]

[QUOTE=Mastan;327850]Ok, Thanks what about this .


2 ^ 2 ^127 -1 which is more clearly POWER(2,(POWER(2,127)) -1[/QUOTE]

2^p-1 iff p is prime, however the reverse p is prime iff 2^p-1 is prime is false, first counter example is 2^11-1 =23*89.

[QUOTE=Mastan;327850]Ok, Thanks what about this .

2 ^ 2 ^127 -1 which is more clearly POWER(2,(POWER(2,127)) -1[/QUOTE]

Sigh... 2 ^ 127 isn't prime. By definition.

Therefore 2 ^ 2 ^ 127 -1 isn't prime.

MM127
 

[QUOTE=Mastan;327850]2 ^ 2 ^127 -1 which is more clearly POWER(2,(POWER(2,127)) -1[/QUOTE]

I think you meant 2^(2^127-1)-1 ?

That's a so called double mersenne number. Its increadibly large and has so far not been shown to be composite.

But I am working on it :smile::

[URL]http://www.doublemersennes.org/mm127.php[/URL]

Hi ,
POWER(2,2)-1 is primenumber (3)
POWER(2,3)-1 is a primenumber (7)
POWER(2,7)-1 is a primenumber (127)
POWER(2,127)-1 is also a primenumber ( 170141183…884105727 )

so this also probably be a primenumber
POWER(2,170141183…884105727 ) -1 . Please let me know if it is really proved to be not a prime number ?

[QUOTE=aketilander;327855]Its increadibly large and has so far not been shown to be composite.

But I am working on it :smile::[/QUOTE]

and I'm rooting for you hard. You've got about a 1% chance of finding a factor with a lot more effort.

if it is a primenumber , can we conclude that 2 POWER (mersenne prime number) is always a prime number, so then we can easily figure out the next largest primenumbers .

[QUOTE=Mastan;327861]if it is a primenumber , can we conclude that 2 POWER (mersenne prime number) is always a prime number, so then we can easily figure out the next largest primenumbers .[/QUOTE]

An example of one of the many downsides of the "Slashdot effect"....

Yes, Aketilander ,Thanks...

Sorry , My question may be misunderstood for someone , my question in tcl scripting clearly is

set x 3
while { $x > 0} {
set x [expr POWER(2,$x) -1]
puts "$x is a prime number ? "
}

Hi Aketilander,

Double mersenne number is only 2 levels of POWER of 2.
I am thinking for more than 2 (may be infinite levels) , it is always a prime number
MMp : Double mersenne number...
MMMMM...p is always a prime number ?


Regards,
Mastan

[QUOTE=Mastan;327864]Sorry , My question may be misunderstood for someone , my question in tcl scripting clearly is

set x 3
while { $x > 0} {
set x [expr POWER(2,$x) -1]
puts "$x is a prime number ? "
}[/QUOTE]

No. 11 is prime, but 2^11-1 is not prime (=23*89, and yes I did have those factors memorized).

[quote=Wikipedia/Mersenne prime]Though it was believed by early mathematicians that Mp is prime for all primes p, [B]Mp is very rarely prime[/B]. In fact, of the [B]1,622,441 prime numbers p[/B] up to 25,964,951,[5] Mp is prime for only [B]42[/B] of them. The smallest counterexample is the Mersenne number
M11 = 211 − 1 = 2047 = 23 × 89.[/quote]

Looking at your Tcl, it looks like you are asking the following, much more specific question(s):
2^2-1 = M2 = 3 is prime.
2^3-1 = M3 = MM2 = 7 is prime.
2^7-1 = M7 = MMM2 = 127 is prime.
2^127-1 = M127 = MMMM2 is prime.

[B]Is MM127 = MMMMM2 a prime?[/B]
For a natural number n and prime p, define M(p, n) = the nth iterated Mersenne number starting with p, such that M(p, 3) = MMMp.
[B]Is M(2, n) prime for all n >= 1?[/B]

There is (as yet) no reason to believe that these numbers have a better chance to be prime than a "random" Mersenne number with a prime exponent. Since Mersenne primes are so few and far between, it is conjectured that MM127 (and M(2, n) for "almost all" n >4) are composite. With our current knowledge of mathematics and computational power though, we have no way to prove it one way or the other. aketilander is looking for a factor of MM127, which would undoubtedly prove its compositeness.

There has been much discussion of these questions -- try Googling "double mersenne numbers" or "iterated mersenne numbers".

[QUOTE=science_man_88;327853]2^p-1 iff p is prime, however the reverse p is prime iff 2^p-1 is prime is false, first counter example is 2^11-1 =23*89.[/QUOTE]

Watch your 'f's there. You're saying contradictory things if those double 'f's are to be interpreted "properly".

[QUOTE=Mastan;327865]
I am thinking for more than 2 (may be infinite levels) , it is always a prime number[/QUOTE]

You may think it, and we cannot disprove it, but there is little reason to believe it is true. In fact it is overwhelmingly believed by math experts to be false.

[QUOTE=Mastan;327865]I am thinking for more than 2 (may be infinite levels) , it is always a prime number
MMp : Double mersenne number...
MMMMM...p is always a prime number ?[/QUOTE]

It will take [I]one[/I] factor to destroy this chain (proving MM127 composite and all of the rest of the chain), and this factor will in a few years be found (most experts agree).

[QUOTE=Dubslow;327877]No. 11 is prime, but 2^11-1 is not prime (=23*89, and yes I did have those factors memorized).



Looking at your Tcl, it looks like you are asking the following, much more specific question(s):
2^2-1 = M2 = 3 is prime.
2^3-1 = M3 = MM2 = 7 is prime.
2^7-1 = M7 = MMM2 = 127 is prime.
2^127-1 = M127 = MMMM2 is prime.

[B]Is MM127 = MMMMM2 a prime?[/B]
For a natural number n and prime p, define M(p, n) = the nth iterated Mersenne number starting with p, such that M(p, 3) = MMMp.
[B]Is M(2, n) prime for all n >= 1?[/B]

There is (as yet) no reason to believe that these numbers have a better chance to be prime than a "random" Mersenne number with a prime exponent. Since Mersenne primes are so few and far between, it is conjectured that MM127 (and M(2, n) for "almost all" n >4) are composite. With our current knowledge of mathematics and computational power though, we have no way to prove it one way or the other. aketilander is looking for a factor of MM127, which would undoubtedly prove its compositeness.

There has been much discussion of these questions -- try Googling "double mersenne numbers" or "iterated mersenne numbers".



Watch your 'f's there. You're saying contradictory things if those double 'f's are to be interpreted "properly".[/QUOTE]

I did I think you need to reread.

[QUOTE=Mastan;327850]Ok, Thanks what about this .


2 ^ 2 ^127 -1 which is more clearly POWER(2,(POWER(2,127)) -1[/QUOTE]

Huh?? It's the difference of two squares! (x^2-y^2) = (x-y)(x+y)..........

[QUOTE=Batalov;327893]It will take [I]one[/I] factor to destroy this chain (proving MM127 composite and all of the rest of the chain)[/quote]

True.

[quote] and this factor will in a few years be found (most experts agree).[/QUOTE]

False.

Hey, George, I'll bet you never contemplated using Tcl or Python to implement your LL software, you dinosaur. No wonder it took so long to find the latest M-prime! Sheesh...

I used 'a few' very, very generously.
I agree that it could be ...erhhhm... a while! (Found a better meaningless term! :-)

[QUOTE=ewmayer;327918]Hey, George, I'll bet you never contemplated using Tcl or Python to implement your LL software, you dinosaur. No wonder it took so long to find the latest M-prime! Sheesh...[/QUOTE]
Nobody uses one-layered software anymore, Ernst! You are behind times.

Definitely Ruby-on-rails. That's it. Must be Ruby on rails. Everybody wears this now.

[QUOTE=Batalov;327920]Everybody wears this now.[/QUOTE]

That's SO yesterhour.

Please see the MersenneWiki:
[url]http://mersennewiki.org/index.php/Double_Mersenne_number[/url]

MM127
 

[QUOTE=Mastan;327865]MMMMM...p is always a prime number ?
[/QUOTE]

No, I don't think so. My guess is that MM127 has about 80 factors, but since its such a large number even the smallest one is very hard to find and require so much work. If I continue searching for another half year, with the speed I have now, I have about 1% chance of finding the smallest one as prime95 said.

For 4 MMp:s factors have already been found, that is for MM13, MM17, MM19, MM31, but maybe you meant that "MMMMM...p [B]for p=2[/B] is always a prime number?", but as I said I believe MM127 has many factors. :smile:

[QUOTE=Prime95;327859]and I'm rooting for you hard. You've got about a 1% chance of finding a factor with a lot more effort.[/QUOTE]

I didn't reply to this George because I was unsure of the meaning of "I'm rooting for you hard". Sometimes english is a little difficult for me, but if it means what I think it means, that is, that you have put a lot of work into programming [URL="http://www.doublemersennes.org/download.php"]mmff[/URL] which I am using in the search, I want to say that I am very grateful for this!!! :smile:

... and who knows maybe we are lucky and find a factor!

[QUOTE=aketilander;328052]I was unsure of the meaning of "I'm rooting for you hard".[/QUOTE]

It means I sincerely hope you find a factor!

[QUOTE=Prime95;328073]It means I sincerely hope you find a factor![/QUOTE]


OK, I see, thanks!!! :bow:

[QUOTE=aketilander;328052]I didn't reply to this George because I was unsure of the meaning of "I'm rooting for you hard". Sometimes english is a little difficult for me, but if it means what I think it means, that is, that you have put a lot of work into programming [URL="http://www.doublemersennes.org/download.php"]mmff[/URL] which I am using in the search, I want to say that I am very grateful for this!!! :smile:

... and who knows maybe we are lucky and find a factor![/QUOTE]You're wise to be cautious. "Rooting" in Australian English has a colloquial meaning. Roughly speaking, that meaning is equivalent to "having sex with an animal".

[QUOTE=xilman;328101]You're wise to be cautious. "Rooting" in Australian English has a colloquial meaning. Roughly speaking, that meaning is equivalent to "having sex with an animal".[/QUOTE]

Now I need to learn the etymology of "rooting"... Does it come from root --> caring form the root --> ?

Luigi

[QUOTE=xilman;328101]You're wise to be cautious. "Rooting" in Australian English has a colloquial meaning. Roughly speaking, that meaning is equivalent to "having sex with an animal".[/QUOTE]

And in Unix geek-speak, "rooting" means to take super-user control (of a system).... :wink:

[QUOTE=ET_;328102]Now I need to learn the etymology of "rooting"...[/QUOTE]

A complete tangent, but anyone who enjoys words would find the book "The Professor and the Madman" (sub-title "A Tale of Murder, Insanity, and the Making of OED") by Esther Lombardi very worth the time.

Another:

root around = rooting, rooted

[url]http://idioms.thefreedictionary.com/root+around[/url]

In US usage (and George's) to root == to cheer. Fans root for their team, etc.

[QUOTE=xilman;328101]You're wise to be cautious. "Rooting" in Australian English has a colloquial meaning. Roughly speaking, that meaning is equivalent to "having sex with an animal".[/QUOTE]

Yeah, it's fun being a kid growing up with an Australian dad, as I did. He'd say stuff like "this is rooted" when something was broken, or "I'm rooted" when he was tired, and then I'd pick up this slang, repeat it in front of my mother, and she'd be all "OMG BBQ Where did you learn that?!?"

I remember hearing it for repeating "hells bells" and "fair suck of the sav" (still not sure what the exact etymology of that one is, but my mother says that there was a sausage variety in Australia known as Savoy, so "fair suck of the sav" could have potentially "dirty" sexual connotations) among others.

Interestingly, I heard "hells bells" (or more like "hay-ells bay-ells") quite a bit whilst living in Georgia. So perhaps it is originally a Scotch/Irish bit of slang that spread more widely in the Empire than one might first expect.

Not as bad as "tabarnak" or "gadzooks" of course...

[QUOTE=NBtarheel_33;328155]Yeah, it's fun being a kid growing up with an Australian dad, as I did. He'd say stuff like "this is rooted" when something was broken, or "I'm rooted" when he was tired, and then I'd pick up this slang, repeat it in front of my mother, and she'd be all "OMG BBQ Where did you learn that?!?"

I remember hearing it for repeating "hells bells" and "fair suck of the sav" (still not sure what the exact etymology of that one is, but my mother says that there was a sausage variety in Australia known as Savoy, so "fair suck of the sav" could have potentially "dirty" sexual connotations) among others.

Interestingly, I heard "hells bells" (or more like "hay-ells bay-ells") quite a bit whilst living in Georgia. So perhaps it is originally a Scotch/Irish bit of slang that spread more widely in the Empire than one might first expect.

Not as bad as "tabarnak" or "gadzooks" of course...[/QUOTE]

rooted --> rotten?

"Hells-bells" --> South of USA had a massive Irish immigration on the last 18th and 19th century. While they usually preferred urban neighborhoods, a good deal of them moved South to help with the estates.

E.T.

[QUOTE=NBtarheel_33;328155]He'd say stuff like "this is rooted" when something was broken, or "I'm rooted" when he was tired[/QUOTE]

Maybe this is a completely different meaning of the word? I come to think of: "[URL="http://en.wikipedia.org/wiki/Bankruptcy#Etymology"]The word [I]bankruptcy[/I] is derived from Italian [I]banca rotta[/I], meaning "broken bench", which may stem from a custom of breaking a moneychanger's bench or counter to signify his insolvency, or which may be only a figure of speech.[/URL]"

Thank you all for a very interesting and clarifying discussion. :smile:

I'm using wordreference.com a lot with "english definition":

[url]http://www.wordreference.com/definition/root[/url]

Last one on that list:

[QUOTE]root
vb
(intransitive) usually followed by for: informal to give support to (a contestant, team, etc), as by cheering
Etymology: 19th Century: perhaps a variant of Scottish rout to make a loud noise, from Old Norse rauta to roar[/QUOTE]

[QUOTE=Mastan;327857]Hi ,
POWER(2,2)-1 is primenumber (3)
POWER(2,3)-1 is a primenumber (7)
POWER(2,7)-1 is a primenumber (127)
POWER(2,127)-1 is also a primenumber ( 170141183…884105727 )

so this also probably be a primenumber
POWER(2,170141183…884105727 ) -1 . Please let me know if it is really proved to be not a prime number ?[/QUOTE]

Using the same reasoning, 3 is prime, 5 is prime, 7 is prime, well, all odd numbers greater than 1 are prime numbers. :smile: Why do you think that the fifth number in the sequence should be prime?

small number law? ;p