Question

Prove that there are no solutions in integers x and y to the equation x^2 − 5y^2 = 2. [Hint: Consider this equation modulo 5.]

Answer 1

To prove that there are no solutions in integers x and y to the equation x^2 - 5y^2 = 2, we can use modular arithmetic. Considering the equation modulo 5, we find that none of the possible residues of x^2 modulo 5 are equivalent to 2. Therefore, there are no solutions to the equation.

Step-by-step explanation:

To prove that there are no solutions in integers x and y to the equation x^2 - 5y^2 = 2, we can use modular arithmetic.

Consider the equation modulo 5:

x^2 - 5y^2 ≡ 2 (mod 5)

Since 5 ≡ 0 (mod 5), the equation can be simplified to:

x^2 ≡ 2 (mod 5)

We can now consider all possible residues of x^2 modulo 5:

• If x ≡ 0 (mod 5), then x^2 ≡ 0 (mod 5). But 0 is not equivalent to 2 (mod 5).
• If x ≡ 1 (mod 5), then x^2 ≡ 1 (mod 5). But 1 is not equivalent to 2 (mod 5).
• If x ≡ 2 (mod 5), then x^2 ≡ 4 (mod 5). But 4 is not equivalent to 2 (mod 5).
• If x ≡ 3 (mod 5), then x^2 ≡ 4 (mod 5). But 4 is not equivalent to 2 (mod 5).
• If x ≡ 4 (mod 5), then x^2 ≡ 1 (mod 5). But 1 is not equivalent to 2 (mod 5).

Therefore, there are no solutions in integers x and y to the equation x^2 - 5y^2 = 2.