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seeking numerical example

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  • Kermit Rose
    (x1^2 + y1^2) (x2^2 + y2^2) = (x1 x2 - y1 y2)^2 + (x1 y2 + x2 y1)^2 = (x1 x2 + y1 y2)^2 + (x1 y2 - x2 y1)^2 expresses the sum of two squares factorization
    Message 1 of 4 , Mar 10 8:10 AM
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      (x1^2 + y1^2) (x2^2 + y2^2) = (x1 x2 - y1 y2)^2 + (x1 y2 + x2 y1)^2 =
      (x1 x2 + y1 y2)^2 + (x1 y2 - x2 y1)^2

      expresses the sum of two squares factorization relationships.

      Suppose we wish to look at the special subset of form { z such that z =
      (t^2 + 1) = p q, where t is integer, and p and q are primes.}

      We can identify elements of that subset by:

      Pick x1,x2,y1,y2 such that x1 x2 - y1 y2 = 1 and
      x1^2 + y1^2 is prime
      and
      x2^2 + y2^2 is prime.

      Is it possible to do this?


      Kermit
    • Maximilian Hasler
      ... depending on the size of the numbers, I think it s faster to consider products of primes and check whether pq-1 is a square. Maximilian
      Message 2 of 4 , Mar 10 9:31 AM
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        On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose <kermit@...> wrote:
        > Suppose we wish to look at the special subset of form { z such that z =
        > (t^2 + 1) = p q, where t is integer, and p and q are primes.}
        >
        > We can identify elements of that subset by:
        >
        > Pick x1,x2,y1,y2 such that x1 x2 - y1 y2 = 1 and
        > x1^2 + y1^2 is prime and x2^2 + y2^2 is prime.
        >
        > Is it possible to do this?


        depending on the size of the numbers,
        I think it's faster to consider products of primes and check whether
        pq-1 is a square.

        Maximilian
      • Maximilian Hasler
        On Sat, Mar 10, 2012 at 1:31 PM, Maximilian Hasler ... forprime(p=1,999,forprime(q=1,p,issquare(p*q-1)&print1(p*q , )))
        Message 3 of 4 , Mar 10 9:36 AM
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          On Sat, Mar 10, 2012 at 1:31 PM, Maximilian Hasler
          <maximilian.hasler@...> wrote:
          > On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose <kermit@...> wrote:
          >> Suppose we wish to look at the special subset of form { z such that z =
          >> (t^2 + 1) = p q, where t is integer, and p and q are primes.}
          >
          > depending on the size of the numbers,
          > I think it's faster to consider products of primes and check whether
          > pq-1 is a square.


          forprime(p=1,999,forprime(q=1,p,issquare(p*q-1)&print1(p*q",")))
          10,26,65,145,82,901,122,2501,2117,1157,485,5777,226,10001,1937,785,6401,362,20737,4097,3601,626,12997,18497,1765,10817,75077,111557,70757,842,64517,2305,81797,2705,7397,266257,1226,37637,23717,254017,11237,3365,320357,9217,144401,448901,1522,3845,207937,276677,60517,270401,527077,244037,104977,38417,712337,698897,

          This (in increasing order) is oeis.org/A144255 : semiprimes of the form n^2+1

          Maximilian
        • Kermit Rose
          ... Given that 10 = 3^2 + 1, I require that x1 x2 - y1 y2 = 1 and x1 y2 + x2 y1 = 3 and x1^2 + y1^2 = 5 and x2^2 + y2^2 = 2 This gives possible solution x1 =
          Message 4 of 4 , Mar 11 6:58 AM
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            On 3/11/2012 9:03 AM, primenumbers@yahoogroups.com wrote:
            > 1c. Re: seeking numerical example
            > Posted by: "Maximilian Hasler"maximilian.hasler@... maximilian_hasler
            > Date: Sat Mar 10, 2012 9:37 am ((PST))
            >
            > On Sat, Mar 10, 2012 at 1:31 PM, Maximilian Hasler
            > <maximilian.hasler@...> wrote:
            >> > On Sat, Mar 10, 2012 at 12:10 PM, Kermit Rose<kermit@...> wrote:
            >>> >> Suppose we wish to look at the special subset of form { z such that z =
            >>> >> (t^2 + 1) = p q, where t is integer, and p and q are primes.}
            >> >
            >> > depending on the size of the numbers,
            >> > I think it's faster to consider products of primes and check whether
            >> > pq-1 is a square.
            > forprime(p=1,999,forprime(q=1,p,issquare(p*q-1)&print1(p*q",")))
            > 10,26,65,145,82,901,122,2501,2117,1157,485,5777,226,10001,1937,785,6401,362,20737,4097,3601,626,12997,18497,1765,10817,75077,111557,70757,842,64517,2305,81797,2705,7397,266257,1226,37637,23717,254017,11237,3365,320357,9217,144401,448901,1522,3845,207937,276677,60517,270401,527077,244037,104977,38417,712337,698897,
            >
            > This (in increasing order) is oeis.org/A144255 : semiprimes of the form n^2+1
            >
            > Maximilian


            Given that 10 = 3^2 + 1,

            I require that

            x1 x2 - y1 y2 = 1

            and

            x1 y2 + x2 y1 = 3

            and
            x1^2 + y1^2 = 5

            and

            x2^2 + y2^2 = 2

            This gives possible solution

            x1 = 2, y1 = 1, x2 = 1, y2 = 1


            Ok. Thank you Max.

            It appears that I have incompletely analyzed the requirements.

            I might or might not get back to you on this problem depending on
            my success in more completely analyzing the requirements of it.

            Kermit
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