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x**2 - y**2 = p**2: Second answer

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  • Kermit Rose
    5. a general formula for the following Posted by: san_tan1 san_tan1@yahoo.co.in san_tan1 Date: Thu Apr 2, 2009 5:15 am ((PDT)) is there any general formula
    Message 1 of 1 , Apr 2, 2009
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      5. a general formula for the following
      Posted by: "san_tan1" san_tan1@... san_tan1
      Date: Thu Apr 2, 2009 5:15 am ((PDT))

      is there any general formula for triples (a,b,c) such that a,b are
      mutually prime and both odd and also a2-b2=c2. ....(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,

      p2= a2-b2=c2-d2=e2-f2.......
      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..?


      Hello san_tan1.


      Given p = 2**b0 *q1**b1 * q2**b2 * q3**b3 . . .
      where each q1,q2,q3,... etc

      are prime,

      then we can list all the ways in which p**2 is the difference
      of two squares.

      Let f be any even divisor of p**2.

      Then p**2 = ( ( p**2 + f)/2 )**2 - ( ( p**2 - f)/2)**2


      Note that in order to make x and y odd,
      f needs to equal 2 mod 4.


      Kermit
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