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Old 2005-07-14, 16:32   #1
R.D. Silverman
 
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Nov 2003

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Question 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.
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Old 2005-07-15, 13:43   #2
dsouza123
 
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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.
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Old 2005-07-15, 14:03   #3
alpertron
 
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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.
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Old 2005-07-15, 14:05   #4
R.D. Silverman
 
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Nov 2003

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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.
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Old 2005-07-15, 14:06   #5
alpertron
 
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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.
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Old 2005-07-16, 23:47   #6
dsouza123
 
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Thank you for the explanation.

Last fiddled with by dsouza123 on 2005-07-16 at 23:49
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Old 2005-07-19, 05:05   #7
antiroach
 
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Jun 2003

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approximately how big are the numbers that would still need factoring in this 'project'?
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