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 2005-07-14, 16:32 #1 R.D. Silverman     Nov 2003 22×5×373 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 state-of-the-art 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.
 2005-07-15, 13:43 #2 dsouza123     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.
 2005-07-15, 14:03 #3 alpertron     Aug 2002 Buenos Aires, Argentina 22×331 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.
2005-07-15, 14:05   #4
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by dsouza123 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.

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, number-theoretic 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.

 2005-07-15, 14:06 #5 alpertron     Aug 2002 Buenos Aires, Argentina 22×331 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.
 2005-07-16, 23:47 #6 dsouza123     Sep 2002 12268 Posts Thank you for the explanation. Last fiddled with by dsouza123 on 2005-07-16 at 23:49
 2005-07-19, 05:05 #7 antiroach     Jun 2003 22×61 Posts approximately how big are the numbers that would still need factoring in this 'project'?

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