But we don't know the general prime numbers

Number Theory Level 1

Find the value of prime numbers $x$ and $y$ satisfying $x^2-2y^2=1.$

x=2,y=3

x=3,y=2

No solution exist

x=y=2

3 solutions

Sandeep Bhardwaj
Nov 22, 2015

However, the problem can be easily solved by verifying solutions method via checking out each option if it satisfies the given equation. But let us try to solve it by general approach:

$x^2-2y^2=1$ gives $x^2=2y^2+1$ . Now the RHS of the equation is an odd number, therefore $x$ must be an odd number. If $x=2n+1$ , then $x^2=(2n+1)^2=4n^2+4n +1=2y^2+1$ .

Therefore $y^2=2n(n+1)$ . This means $y^2$ is an even number and hence $y$ is also an even number. And the only even prime number is 2. Now we can easily conclude that $y=2$ . Putting this in the equation we get $x=3$ .

Hence $x=3,y=2$ is the only solution of the given equation provided $x,y$ are prime numbers. $\square$

Moderator note:

Working modulo 4, we can obtain that $y$ must be even, hence it is equal to 2.

More generally, $x^2 - 2y^2 = 1$ is known as a pell's equation, and the solutions are of the form $(x_n + y_n \sqrt{2} = ( 3 + 2 \sqrt{2})^n$ .

Tushar Kaushik
Nov 26, 2015

Just put the values

I have a another one solution X=√2 Y=1/√2

Bal Samy - 5 years, 6 months ago

That is not a prime. Prime number is defined in terms of natural number (zero and positive integer only).

Kay Xspre - 5 years, 6 months ago

Harish Kumar
Nov 24, 2015

As we know that 3 and 2 both are prime numbers. Hence let if x=3 and y=2 than by putting these in the given equation. 1 is the answer that we get.

Hence x=3 and y=2 both prime numbers satisfied the equation. So these values are the answer.

But we don't know the general prime numbers

3 solutions

Moderator note:

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