20050714, 16:32  #1 
Nov 2003
16444_{8} Posts 
Project proposal?
I will agree with the statement:
"Knowing if M_p 2 is also prime" is somewhat interesting. However, I would like to observe that in the current stateoftheart in factoring, that trying to factor M_p 2 is HOPELESS, except for the smallest p. The level of effort spent so far has not been unreasonable. (IMO). However, I think spending yet more time is unlikely to lead to success(es). Allow me to instead suggest an alternative project which does have some hope of success: Extending the Cunningham project to 'homogeneous form', i.e. numbers of the form A^n  B^n with (A,B) = 1, B>1. [Cunningham is just B = 1] I have already done some modest work in this area. I have completed the following: ("Base x to y" means that I have completed A^n  B^n for A = x and all B < A up to exponent y with (A,B) = 1) Base 3 to 330 Base 4 to 256 Base 5 to 225 Base 6 to 195 Base 7 to 165 Base 8 to 155 Base 9 to 135 Base 10 to 130 Base 11 to 130 Base 12 to 120 I will make these results available to anyone who asks. I don't post them here; the current tables are ~600Kbytes total. Perhaps some of you might like to extend these? Such an effort would be achievable. I am also cross posting this to the ''twin prime' discussion. 
20050715, 13:43  #2 
Sep 2002
2·331 Posts 
What does (A,B)=1 mean in the context of your project proposal ?
For the first item on the list Base 3 to 330 does it evaluate to 3^0  0^0 3^0  0^1 ... 3^0  0^329 3^0  0^330 3^1  1^0 ... 3^1  1^330 ... 3^330  2^0 ... 3^330  2^330 If the preceding is incorrect, then please show the correct evaluation. 
20050715, 14:03  #3 
Aug 2002
Buenos Aires, Argentina
2453_{8} Posts 
It appears that he wants to factor:
3^2  2^2 3^3  2^3 3^4  2^4 ... 3^330  2^330 4^2  2^2 ... 4^256  2^256 4^2  3^2 ... 4^256  3^256 where the exponents are the same. In that case there are forms of Aurifeuillan factorizations that help factor many of these numbers. There is a publication of Richard Brent about these Aurifeuillian factorizations. 
20050715, 14:05  #4  
Nov 2003
2^{2}×5×373 Posts 
Quote:
I wrote: numbers of the form A^n  B^n with (A,B) = 1, B>1. [Cunningham is just B = 1] (A,B) = 1 is standard, numbertheoretic notation. (x,y) is the GCD of x and y. And since I clearly wrote A^n  B^n, I do not understand why you are suggesting putting different exponents on A and B. And since I also clearly wrote B > 1, I do not understand why you are putting B=0. Was my writing unclear???? BTW, I have also done A^n + B^n to the same limits. 

20050715, 14:06  #5 
Aug 2002
Buenos Aires, Argentina
3^{3}·7^{2} Posts 
It appears that I made a mistake.
4^2  2^2 ... 4^256  2^256 should not be included because (A,B)>1. They are already factored in the Cunningham project. 
20050716, 23:47  #6 
Sep 2002
2·331 Posts 
Thank you for the explanation.
Last fiddled with by dsouza123 on 20050716 at 23:49 
20050719, 05:05  #7 
Jun 2003
244_{10} Posts 
approximately how big are the numbers that would still need factoring in this 'project'?

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