11(0 ,-2)
(2, 0)
I dont think there r more.
Find all pairs (x, y) of integers such that x^3 - y^3 = 2xy + 8
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11 Answers
1(2,0)
(0,-2)
even i don't think der r more............
some more r der?
39accord to me alo, more solns wont exist.
but can any of u guys prove it ?????????
11i did putting-patting..no logic actually so cant prove it...thats d way i do all these num. theory type sums
9top class sol
dipanjan
b555 is there a number theory type sol involving wierd identities ?
11Gr8 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)}.
11Some 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.