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a general formula for the following

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  • san_tan1
    is there any general formula for triples (a,b,c) such that a,b are mutually prime and both odd and also a^2-b^2=c^2. ....(1)? also can this be extended to
    Message 1 of 3 , Apr 2, 2009
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      is there any general formula for triples (a,b,c) such that a,b are mutually prime and both odd and also a^2-b^2=c^2. ....(1)?

      also can this be extended to finding an algorithm for generating duplets (a,b) (c,d) etc....such that given p we can find upto any number of desired duplets having property 1??

      that is given p,

      p^2= a^2-b^2=c^2-d^2=e^2-f^2.......
      is there any way to generate (a,b) (c,d) ,(e,f)....upto any desired no. of duplets such that in each dupet(x,y) x,y are both odd and mutually prime..?
    • David Broadhurst
      ... Yes: (4*n^2+1)^2 - (4*n^2-1)^2 = (4*n)^2 ... No: for p = 0 mod 4, the number of odd coprime pairs [x,y] with p = x^2 - y^2 is 2^k where k is the number of
      Message 2 of 3 , Apr 2, 2009
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        --- In primenumbers@yahoogroups.com,
        "san_tan1" <san_tan1@...> wrote:

        > is there any general formula for triples (a,b,c) such that
        > a,b are mutually prime and both odd and also
        > a^2-b^2=c^2

        Yes: (4*n^2+1)^2 - (4*n^2-1)^2 = (4*n)^2

        > given p,
        > p^2= a^2-b^2=c^2-d^2=e^2-f^2.......
        > is there any way to generate (a,b) (c,d) ,(e,f)....up
        > to any desired no. of duplets such that in each duplet(x,y)
        > x,y are both odd and mutually prime

        No: for p = 0 mod 4, the number of odd coprime pairs [x,y]
        with p = x^2 - y^2 is 2^k where k is the number of distinct
        odd prime divisors of p. This procedure will print them all:

        {pairs(p)=local(x,y,c,d);if(p%4==0,fordiv(p/2,d,
        c=p/2/d;if(c>d,x=c^2+d^2;y=c^2-d^2;if(gcd(x,y)==1,
        print([x,y])))));}

        pairs(2^3*3^2*5*7*11);

        [192099601, 192099599]
        [12006241, 12006209]
        [7684009, 7683959]
        [3920449, 3920351]
        [2371681, 2371519]
        [1587721, 1587479]
        [480649, 479849]
        [245809, 244241]
        [158041, 155591]
        [149521, 146929]
        [101161, 97289]
        [96889, 92839]
        [66529, 60479]
        [52369, 44431]
        [38329, 26471]
        [29401, 9799]

        David
      • David Broadhurst
        PS: I left out a square sign, here restored: for p = 0 mod 4, the number of odd coprime pairs [x,y] with p^2 = x^2 - y^2 is 2^k where k is the number of
        Message 3 of 3 , Apr 2, 2009
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          PS: I left out a square sign, here restored:

          for p = 0 mod 4, the number of odd coprime pairs [x,y]
          with p^2 = x^2 - y^2 is 2^k where k is the number of distinct
          odd prime divisors of p.

          David^2
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