Prime conjecture

[Stan]

Let m be a positive odd integer.

If 2^m = 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?

[retina]

I have no idea if it is provable or not, but there are only two more solutions below 10^9:

42463, 71443

[science_man_88]

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.

[retina]

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, ...

[science_man_88]

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.

[Stan]

Originally Posted by Stan

Let m be a positive odd integer.

If 2^m = 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.

[ATH]

nevermind, was not correct.

[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 2^m = 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?

[science_man_88]

Originally Posted by ATH

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

[science_man_88]

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 2^m = 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:

M_m = 1 mod (m(m-1))

we know:

M_m = 1 mod m, if m is prime

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

[ewmayer]

Dude, did you really need to start 3 separate threads on this? I merged all of 'em here.