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 2012-09-16, 09:57 #1 Stan   Dec 2011 22·32 Posts Prime conjecture Let m be a positive odd integer. If 2m = 2 (mod. m(m-1)) then m is a prime. True or False? This is true for m = 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367. These are all primes and there are no other values lessthan 40 003. Is it possible to prove the conjecture?
 2012-09-16, 11:56 #2 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 1A4016 Posts I have no idea if it is provable or not, but there are only two more solutions below 10^9: 42463, 71443
2012-09-16, 11:59   #3
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

100000111011002 Posts

Quote:
 Originally Posted by Stan Let m be a positive odd integer. If 2m = 2 (mod. m(m-1)) then m is a prime. True or False? This is true for m = 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367. These are all primes and there are no other values lessthan 40 003. Is it possible to prove the conjecture?
for these value of m these become:

6 = 2 mod (9-3=6) which is false
14 = 2 mod (49-7=42) which is false
38 = 2 mod (361-19=342) which is false

the main problem is that m(m-1) = m^2-m which is larger than 2m once m>3 so all m>3 fail this test. or at least that's my interpretation of it.

I looked up your phrasing and found it on mathhelpforum.com and apparently it's 2^m not 2m as it reads here.

Last fiddled with by science_man_88 on 2012-09-16 at 12:06

2012-09-16, 13:31   #4
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

26×3×5×7 Posts

Quote:
 Originally Posted by retina I have no idea if it is provable or not, but there are only two more solutions below 10^9: 42463, 71443
Scratch that. My minion's little task turns out to be in error.

There are indeed a lot more solutions for that up to 10^9.

Here are the first few (all prime): 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827, 318403, 351919, 370387, 388963, 524287, 543607, 554527, 581743, 585847, 674083, 688087, 698923, ...

Last fiddled with by retina on 2012-09-16 at 13:36

2012-09-16, 13:40   #5
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

22×72×43 Posts

Quote:
 Originally Posted by retina Scratch that. My minion's little task turns out to be in error. There are indeed a lot more solutions for that up to 10^9. Here are the first few (all prime): 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827, 318403, 351919, 370387, 388963, 524287, 543607, 554527, 581743, 585847, 674083, 688087, 698923, ...
They posted again in homework help it looks like.

2012-09-16, 14:28   #6
Stan

Dec 2011

448 Posts
Prime conjecture

Quote:
 Originally Posted by Stan Let m be a positive odd integer. If 2m = 2 (mod. m(m-1)) then m is a prime. True or False? This is true for m = 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367. These are all primes and there are no other values lessthan 40 003. Is it possible to prove the conjecture?
I hope the above is now correct.

 2012-09-16, 14:56 #7 ATH Einyen     Dec 2003 Denmark 23·431 Posts nevermind, was not correct. Last fiddled with by ATH on 2012-09-16 at 15:06
 2012-09-16, 15:05 #8 Stan   Dec 2011 22·32 Posts Mersenne prime sequence If it is possible to prove the following conjecture then I can prove the existance of an infinite sequence of Mersenne primes and I can represent them as an algebraic expression. However, they are not representable as numbers. Let m be a positive odd integer. If 2m = 2 (mod. m(m-1)) then m is a prime. True or False? This is true for m = 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, and these are all prime. I can find no composite integers for which the statement holds. Is it possible to prove the conjecture? Last fiddled with by Stan on 2012-09-16 at 15:09 Reason: Add icon
2012-09-16, 15:28   #9
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

20EC16 Posts

Quote:
 Originally Posted by ATH nevermind, was not correct.

and probably more.

oh yeah I forgot I originally found it correctly on:

http://mathhelpforum.com/number-theo...onjecture.html

Last fiddled with by science_man_88 on 2012-09-16 at 15:31

2012-09-16, 16:15   #10
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

100000111011002 Posts

Quote:
 Originally Posted by Stan If it is possible to prove the following conjecture then I can prove the existance of an infinite sequence of Mersenne primes and I can represent them as an algebraic expression. However, they are not representable as numbers. Let m be a positive odd integer. If 2m = 2 (mod. m(m-1)) then m is a prime. True or False? This is true for m = 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, and these are all prime. I can find no composite integers for which the statement holds. Is it possible to prove the conjecture?
well reformatting gets to:

Mm = 1 mod (m(m-1))

we know:

Mm = 1 mod m, if m is prime

so Mm would need the correct residue mod (m-1) for this to work.

 2012-09-16, 20:09 #11 ewmayer ∂2ω=0     Sep 2002 República de California 5×2,351 Posts Dude, did you really need to start 3 separate threads on this? I merged all of 'em here.

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