crack this

(2,0)
(0,-2)

are there some more ?

(2,0)
(0,-2)
even i don't think der r more............
some more r der?

accord to me alo, more solns wont exist.

but can any of u guys prove it ?????????

i did putting-patting..no logic actually so cant prove it...thats d way i do all these num. theory type sums

top class sol
dipanjan

b555 is there a number theory type sol involving wierd identities ?

Gr8 stuff Jishnu.

Alternately, let us have 2 cases, where (x,y)=q ,i.e x and y are not relatively prime integers.

Then we get the 2 solutions of q as 2 or 1 (the trivial case).

Taking q=2, we have

x-y=a and x+y=b, so our en at hand gets transformed to

a(b+a)^2+a(b-a)^2+a(b^2-a^2)-(b^2-a^2)=4 from which we have

b^2(3a-1)=4+a^2-a^3.

Taking a>0, we have the only soln. of a as 1.

Next we may have a<0, in which case, we can havbe y-x=a and y+x as b. Similar soln follows.

Now lets deal with the case when (x,y)=1.

With similar substitutions as before, we have b^2(3a-2)=-(a^3+2a^2-32).

For a>0, we must have r.h.s>0. Which gives a=(1,2).

Figuring out consequent values of b we have the soln. set of the eqn as
\boxed{(x,y)=(2,0),(0,-2)}.

Some typos btw :- In #5, x and y are not the x and y's in the eqn.

Here x=r and y=p, such that the original x and y's are x=qr and y=qp.

Rest seems ok.