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 Register FAQ Search Today's Posts Mark Forums Read  2010-12-15, 15:16 #1 kurtulmehtap   Sep 2009 22·32 Posts Properties of Mersenne numbers In Tony Reix's Properties of Mersenne and Fermat numbers online paper you see: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2. The proof is missing. Can anybody provide a proof? By numerical testing different Mq values I have found that if Mq is composite there is no pair (x,y) that satisfies the condition. Is it possible that if Mq is composite there can be 2 or more pairs? Thanx in advance...   2010-12-15, 15:35   #2
R.D. Silverman

Nov 2003

22×5×373 Posts Quote:
 Originally Posted by kurtulmehtap In Tony Reix's Properties of Mersenne and Fermat numbers online paper you see: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2. The proof is missing. Can anybody provide a proof?
I have not verified that the result is true. I will assume that it is.

I will sketch a proof. This result has very little to do with Mersenne
primes.

Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique.

Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime
iff it is the sum of two squares in a unique way. i.e. --Factor Q over
Q(sqrt(-3)) and observe that you are doing so in a UFD.

[QUOTE]   2010-12-15, 21:20   #3
davar55

May 2004
New York City

5×7×112 Posts Quote:
 Originally Posted by R.D. Silverman I have not verified that the result is true. I will assume that it is. I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. --Factor Q over Q(sqrt(-3)) and observe that you are doing so in a UFD.
Pretty easy, huh? Good job.   2010-12-16, 16:06   #4
kurtulmehtap

Sep 2009

1001002 Posts Quote:
 Originally Posted by R.D. Silverman I have not verified that the result is true. I will assume that it is. I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. --Factor Q over Q(sqrt(-3)) and observe that you are doing so in a UFD.
Thanks a lot.
I'm stuck. If there is a unique pair (x,y) then Q is prime,however , if Q is composite, then can we assume that there are no (x,y) pairs or should we consider there are 2,3 or more pairs?
Thanks

Last fiddled with by wblipp on 2010-12-16 at 19:08 Reason: fix quotes   2010-12-16, 18:43   #5
R.D. Silverman

Nov 2003

22×5×373 Posts [QUOTE=kurtulmehtap;242188]
Quote:
 Originally Posted by R.D. Silverman I have not verified that the result is true. I will assume that it is. I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. --Factor Q over Q(sqrt(-3)) and observe that you are doing so in a UFD. Thanks a lot. I'm stuck. If there is a unique pair (x,y) then Q is prime,however , if Q is composite, then can we assume that there are no (x,y) pairs or should we consider there are 2,3 or more pairs? Thanks   2010-12-17, 19:41   #6
davar55

May 2004
New York City

423510 Posts Quote:
 Originally Posted by R.D. Silverman Hint: Composition of quadratic forms...
Another hint: think third degree polynomial equations over Z.   2010-12-23, 21:48 #7 davar55   May 2004 New York City 5×7×112 Posts So is the OPer satisfied?   2011-01-05, 14:15   #8
kurtulmehtap

Sep 2009

22×32 Posts Quote:
 Originally Posted by davar55 So is the OPer satisfied?
Not Really, I am still not sure if a composite Mersenne number can have more than 1 pair for x^2 + 27y^2.

There is a thesis on this subject:
Mersenne primes of the form x^2+dy^2 by Bas Jansen at
www.math.leidenuniv.nl/en/theses/31/

It has an entire section for the needed case d=27, but I still can't find the answer..

I know that I am embarassing myself but I need the answer.   2011-01-05, 15:20   #9
R.D. Silverman

Nov 2003

164448 Posts Quote:
 Originally Posted by kurtulmehtap Not Really, I am still not sure if a composite Mersenne number can have more than 1 pair for x^2 + 27y^2. There is a thesis on this subject: Mersenne primes of the form x^2+dy^2 by Bas Jansen at www.math.leidenuniv.nl/en/theses/31/ It has an entire section for the needed case d=27, but I still can't find the answer.. Please help. I know that I am embarassing myself but I need the answer.
If the number is composite, there will be more than 1.   2011-01-05, 15:22   #10
R.D. Silverman

Nov 2003

1D2416 Posts Quote:
 Originally Posted by R.D. Silverman If the number is composite, there will be more than 1.
Look up "idoneal".   2011-01-05, 15:23   #11
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts Quote:
 Originally Posted by R.D. Silverman Look up "idoneal".   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post princeps Miscellaneous Math 18 2011-11-30 00:16 mack Information & Answers 2 2009-09-07 05:48 henryzz Lounge 4 2008-11-30 20:46 henryzz Math 2 2008-04-29 02:05 T.Rex Math 4 2005-05-07 08:25

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