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Re: really? (lame question about primes)

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On 02.09.2015 16:37, David Brown wrote:
> On 02/09/15 16:27, bartekltg wrote:
>> On 02.09.2015 15:47, David Brown wrote:
>>> On 02/09/15 14:56, bartekltg wrote:
>>>> On 02.09.2015 14:19, Richard Heathfield wrote:
>>>>> On 02/09/2015 13:12, David Brown wrote:
>>>>>>
>>>>> <snip>
>>>>>>
>>>>>> There are a number of ways of doing this check faster than the
>>>>>> bit-for-bit long division you suggested elsewhere,
>>>>>
>>>>> The advantage of that way is that it's easy to explain, and relatively
>>>>> easy to implement. Having said that, I *haven't* implemented it (in
>>>>> software). My bignum library uses Algorithm D, from TAOCP II.
>>>>
>>>> I think explain something by using wrong algorithm can hurt
>>>> someone who think, so propably everyone who is listening to
>>>> 'how to factorize an integer;-)'
>>>>
>>>> But we can resurect this idea.
>>>>
>>>> Table of all n-digits prime can be too big for the universe,
>>>> so we have to generate it. One ot the simplest method is
>>>> to implement 'nextprime' - function that get the next odd*)
>>>> number and run fast primality test (Miller-Rabin),
>>>> repeat it until the number is pseudoprime.
>>>>
>>>
>>> The principle here is that to find the factors of C, we run through all
>>> primes A up to sqrt(C), and test if C % A == 0.  (It may be that there
>>> are other faster algorithms, but I don't know of any.)  If you know for
>>> sure the number of digits in A, then you could start 10^(noOfDigits-1),
>>> but that would only save 10% of the test time.
>>>
>>> Testing for the primality of each A is going to take a great deal more
>>> time than simply trying C % A == 0, even with a probable prime test.  It
>>> is simply not worth doing.
>>>
>>> What would be worth doing, I think, is to keep a track of A % p for p in
>>> the first n primes.  As you run through the A's, adding 2 to A each
>>> step, you add 2 to each of these remainders in your tracking array.  If
>>> these hit 0 (modulo their prime), then A is clearly non-prime and you
>>> can skip it.  This will let you skip a number of your test divisions
>>> with relatively little work - as you make that array of n primes bigger,
>>> you cut out yet more divisions.  However, there is going to be a point
>>> where doing this tracking for the modulo A % p will cost more (in memory
>>> or speed) than 1/p times the cost of calculating C % A - and when you
>>> reach that, you don't want to track any more modulos.
>>>
>>> I suspect there is scope for some interesting data structures to hold
>>> this list of modulos, rather than a simple array - perhaps a priority
>>> queue in a Fibonacci heap?
>>
>>
>> Yes, this is the "standard" algorithm I was talking about, in the last
>> post I even mentioned the heuristic you described it the third
>> paragraph.
>> Are you sure you wanted to post it as the answer to my post?
>>
>
> That's not how I understood your post.  Though I agree the basic
> algorithm is the same, the details are very different.  In particular,

I did not describe the 'standard' algorithm but explore Richarsd's/fir's
idea about an algorithm without divisions, but writt also without
checking all pairs.

> your third paragraph suggested Miller-Rabin testing before division,
> which I think would be terribly inefficient.

Yes, you are right, we do not have to check if A is (pseudo)prime,
and in reality, like you said, M-R test is slower than division.

This was written that way to resemble 'check all pair',
but yes, it was stupid idea;-)

bartekltg

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really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-08-31 12:07 -0700
  Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-08-31 23:23 +0100
    Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-01 01:18 -0700
      Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-01 11:01 +0200
        Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-01 12:20 +0100
          Re: really? (lame question about primes) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2015-09-01 15:48 +0100
            Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-01 13:26 -0700
              Re: really? (lame question about primes) Robert Wessel <robertwessel2@yahoo.com> - 2015-09-02 02:12 -0500
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 08:38 +0100
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-02 14:03 +0200
          Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-01 13:24 -0700
            Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-01 22:27 +0100
              Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-01 15:02 -0700
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-01 23:20 +0100
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 01:22 +0200
                Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-01 17:18 -0700
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 02:19 +0100
                Re: really? (lame question about primes) James Dow Allen <gmail@jamesdowallen.nospam> - 2015-09-02 12:42 +0000
                Re: really? (lame question about primes) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2015-09-02 14:21 +0100
                Re: really? (lame question about primes) Johannes Bauer <dfnsonfsduifb@gmx.de> - 2015-09-02 17:45 +0200
                Re: really? (lame question about primes) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2015-09-02 18:06 +0100
                Re: really? (lame question about primes) James Dow Allen <jdallen2000@yahoo.com> - 2015-09-03 17:05 -0700
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-04 02:18 +0200
                Re: really? (lame question about primes) Chad <cdalten@gmail.com> - 2015-09-03 22:37 -0700
                Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-03 23:49 -0700
                Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-04 00:18 -0700
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-04 08:25 +0100
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 14:36 +0100
                Re: really? (lame question about primes) Ben Bacarisse <ben.usenet@bsb.me.uk> - 2015-09-02 16:49 +0100
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 17:32 +0100
            Re: really? (lame question about primes) Kaz Kylheku <kaz@kylheku.com> - 2015-09-01 22:55 +0000
              Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 01:15 +0200
          Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 03:19 +0200
            Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 02:37 +0100
              Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 11:24 +0200
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 10:42 +0100
                Re: really? (lame question about primes) fir <profesor.fir@gmail.com> - 2015-09-02 03:01 -0700
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 12:28 +0200
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-02 14:12 +0200
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 13:19 +0100
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 14:56 +0200
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-02 15:47 +0200
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 16:27 +0200
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-02 16:37 +0200
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-02 17:01 +0200
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-03 10:55 +0200
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-03 10:05 +0100
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-03 11:49 +0200
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-03 10:54 +0100
                Re: really? (lame question about primes) David Brown <david.brown@hesbynett.no> - 2015-09-03 13:22 +0200
                Re: really? (lame question about primes) bartekltg <bartek@gmail.com> - 2015-09-03 13:58 +0200
                Re: really? (lame question about primes) Richard Heathfield <rjh@cpax.org.uk> - 2015-09-02 13:34 +0100
  Re: really? (lame question about primes) Robert Wessel <robertwessel2@yahoo.com> - 2015-08-31 23:10 -0500

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