x**2 - y**2 = p**2: Second answer
- 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,
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
Given p = 2**b0 *q1**b1 * q2**b2 * q3**b3 . . .
where each q1,q2,q3,... etc
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.