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Old 2012-09-16, 09:57   #1
Stan
 
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Question 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?
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Old 2012-09-16, 11:56   #2
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I have no idea if it is provable or not, but there are only two more solutions below 10^9:

42463, 71443
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Old 2012-09-16, 11:59   #3
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Quote:
Originally Posted by Stan View Post
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
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Old 2012-09-16, 13:31   #4
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Quote:
Originally Posted by retina View Post
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
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Old 2012-09-16, 13:40   #5
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Quote:
Originally Posted by retina View Post
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.
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Old 2012-09-16, 14:28   #6
Stan
 
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Default Prime conjecture

Quote:
Originally Posted by Stan View Post
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.
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Old 2012-09-16, 14:56   #7
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nevermind, was not correct.

Last fiddled with by ATH on 2012-09-16 at 15:06
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Old 2012-09-16, 15:05   #8
Stan
 
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Lightbulb 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
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Old 2012-09-16, 15:28   #9
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Quote:
Originally Posted by ATH View Post
nevermind, was not correct.
see also:

http://mersenneforum.org/showthread.php?p=311833
http://mersenneforum.org/showthread.php?p=311851

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
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Old 2012-09-16, 16:15   #10
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Quote:
Originally Posted by Stan View Post
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.
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Old 2012-09-16, 20:09   #11
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Dude, did you really need to start 3 separate threads on this? I merged all of 'em here.
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