20110105, 19:40  #12 
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 

20110105, 19:45  #13 
Sep 2009
100100_{2} Posts 

20110105, 23:08  #14 
Einyen
Dec 2003
Denmark
3,313 Posts 
Theorem checks out up to p=73, though thats not very far. Mersenne primes have 1 solution and composite numbers have 0 or 2:
Code:
p (x,y) so 2^{p}1=4*x^{2}+27*y^{2} 5 1,1 7 4,1 11 no solution 13 23,15 17 181,1 19 149,127 23 no solution 29 no solution 31 23081,783 37 142357,45695 and 185341,1119 41: no solution 43: no solution 47: no solution 53: no solution 59: no solution 61: 752652049,38443119 67: 4922679991,1369547633 and 5053371809,1297114833 71: no solution 73: no solution Last fiddled with by ATH on 20110105 at 23:11 
20110106, 15:55  #15 
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
I wish finding primes in lucas sequences were easy lol, if so we could rely on the fact that mersenne numbers are U(3,2) if I remember correctly.
Last fiddled with by science_man_88 on 20110106 at 16:06 
20110107, 13:35  #16 
"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts 
(2x)^2+ 3(3y)^2 could be transformed to:
(Qx)^2 + P(Py)^2 which can technically at least in this case be transformed to: (Qx)^Q + P(Py)^Q which may be transformed further I believe but I can't remember enough math right now to do that anyone else up to looking at this ? 
20110107, 14:53  #17  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
Last fiddled with by science_man_88 on 20110107 at 15:01 

20110107, 15:19  #18 
Einyen
Dec 2003
Denmark
CF1_{16} Posts 
The conjecture is: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2.
So you can not make the 2 and 3 into new variables P and Q. Then its not this conjecture anymore, and if P and Q can be anything, then the conjecture is most likely not true anymore, since there is probably more solutions for different P and Q. Last fiddled with by ATH on 20110107 at 15:20 
20110107, 15:24  #19  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
Last fiddled with by science_man_88 on 20110107 at 15:50 

20110107, 15:52  #20 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
I see where I went wrong above in my expansion I multiplied by p in 2 places not one doh.

20110107, 20:24  #21  
"Forget I exist"
Jul 2009
Dumbassville
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Quote:


20110108, 03:12  #22 
Aug 2006
3×1,993 Posts 

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