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Re: [PrimeNumbers] Re: Cross Base Factorising

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  • Kevin Acres
    Actually it is practical. Remember that you don t store the actual number, only the exponent. So in the example of 3^88-1 I know that everything below my
    Message 1 of 6 , Jul 1, 2004
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      Actually it is practical.

      Remember that you don't store the actual number, only the exponent. So in
      the example of 3^88-1 I know that everything below my addition point is
      2222222..... As such it gets discarded. I only count how many 2's I get in
      the string.

      Kevin.




      At 04:26 PM 2/07/2004, eharsh82 wrote:
      >Yes it should work!
      >
      >"Take a number known to be composite, in this case 2047 and check it
      >against the smallest 3^n -1 number that it can divide."
      >
      >Finding n is really difficult for large numbers. So your algorithm is
      >not practical.
      >
      >
      >Harsh
      >
      >
      >
      >
      >
      >--- In primenumbers@yahoogroups.com, Kevin Acres <research@r...>
      >wrote:
      > > OK, here's a bombproof way to factorize a Mersenne, given that the
      > > algorithm is implementable in a computer. Please make me aware of
      >any
      > > counter-example should you know of one.
      > >
      > > Remember that we are dealing solely with an exponent of a number to
      >a given
      > > base. You don't need to keep the actual number itself in computer
      >memory
      > > in an implementation.
      > >
      > > Given that the "reverse method" can find the smallest exponent in a
      >given
      > > base for a given number.
      > >
      > > And given that p divides (n^(p-1))-1 where p is prime or a base n
      > > pseudoprime. Taken another way p+1 divides n^p - 1.
      > >
      > > Take a number known to be composite, in this case 2047 and check it
      >against
      > > the smallest 3^n -1 number that it can divide.
      > >
      > > The reverse method shows us that 2047 divides 2^11-1 (itself) and
      >3^88 - 1.
      > >
      > > We now know for sure that 2047 isn't prime because 88 doesn't
      >divide 2046,
      > > (2047-1).
      > >
      > > We also know that x+1 will divide 3^x -1. In this case x+1 is 88+1
      >or 89.
      > >
      > > Now we find that 89 divides 2047 giving us 23, the other factor.
      > > Effectively factorizing 2^11-1 without ever having to work in base
      >2.
      > >
      > > Now take another number, this time known to be prime, 8191
      > >
      > > Applying the reverse algorithm in base 3 we find that 8191 divides
      >3^8190 -
      > > 1. 8190 divides 8191-1 so we now that 8191 is prime in base 3. The
      >reverse
      > > algorithm in base 2 shows that 8191 divides 2^13-1 and that 13
      >divides
      > > 8190, therefore 8191 is base 2 prime as well.
      > >
      > > So am I wrong or does this method of factorizing work in all cases
      >for
      > > Mersenne numbers?
      > >
      > > Kevin.
      >
      >
      >
      >
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      >The Prime Pages : http://www.primepages.org/
      >
      >
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      >
      >
      >
      >
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