Guest

Prove that x^2-y^2=2 has no integral solutions.

Prove that x^2-y^2=2 has no integral solutions.

Grade:

3 Answers

Ankit Khokhar
21 Points
10 years ago

Ok lets see, write the equation as:

x^2 = y^2 + 2

Now, so

x = (y^2 + 2)^(1/2)

Now the above equation represents a hyperbola ( by general form: (x^2)/(a^2) - (y^2)/(b^2) =1 )

so, it will be symmetric,

so if it have any integral coordiantes ( or any integral solution of above equation in any of the four quadrants then it should have in all four , and if not in any of one then in none of them )

so lets take out the first quadrant,

here x and y, both are positive,

Now vary the value of y so that we can get corresponding values of x (and can see if both be integers)

lets start by least integer (considering the first quadrant) which is zero, 0

so if y = 0,

x=(2)^(1/2)=1.414 , not an integer

now let y = 1, so x=(3)^(1/2) = 1.732,not an integer again

now let y=2, so x=(4 + 2)^(1/2) = 6^(1/2), again not an integer

Now from 2 and above, the perfect squares will be at a distance more than 2, so cannot get a perfect square by adding a perfect square with 2

REASON: we have to get:    Perfect square = another Perfect square + 2

Now for 2, square = 2*2

for 3 square = 3*3,

so the distance between consecutive square increases as the number increases,

so no integral solutions can be obtained afterwards,

Best of Luck for your exams

Prasenjit Dubey
17 Points
10 years ago

(x+y)(x-y)=2

2 is a prime no.

hence either x+y=2 x-y=1

               or vice versa

in either case not posible

Adhiraj Mandal
32 Points
10 years ago

Let us assume that x^2-y^2=2 has a integral solution.

Then (x+y)(x-y)=2

the only way 2 can be factorized is 2 = 2*1

As x+y>x-y we can assume that x+y=2 ....................... 1 and

                                              x-y=1.........................2

From 1   x=2-y  and from 2   x = 1+y

therefore y is an integer between 1 and 2.....................this is a contradiction to our assumption.

Hence our assumption must be wrong.

Hence x^2-y^2=2 has no integral solution.  [HENCE PROVED].

 

Please approve my solution in case you like it.

 

                                                                  ADHIRAJ MANDAL

Think You Can Provide A Better Answer ?

ASK QUESTION

Get your questions answered by the expert for free