Given x^2 = y + a and y^2 = x + a where a is a positive integer, find expressions for that yield integer solution for x and y.
To find expressions that yield integer solutions for x and y in the given equations, we can simplify them and try to manipulate them to form perfect squares.
Starting with the first equation: x^2 = y + a
Let's subtract 'a' from both sides: x^2 - a = y
From this equation, we see that for x and y to be integers, x^2 must be congruent to a (mod a), which means x^2 must leave a remainder of 'a' when divided by 'a'. In other words, x^2 ≡ a (mod a).
Now, let's look at the second equation: y^2 = x + a
Again, subtract 'a' from both sides: y^2 - a = x
Similar to the first equation, for y and x to be integers, y^2 must be congruent to 'a' (mod a), which means y^2 ≡ a (mod a).
To further simplify the expressions, we can find the prime factorization of 'a'. Let's say 'a' can be written as a = p₁ * p₂ * p₃ * ... * pₙ, where p₁, p₂, ..., pₙ are distinct prime numbers.
Now, let's consider the congruences x^2 ≡ a (mod pᵢ) and y^2 ≡ a (mod pᵢ) for each prime factor pᵢ of 'a'. By the Chinese Remainder Theorem, if we find a solution for each prime factor, we can find a solution for the entire equation.
For each prime factor pᵢ, we need to find integers x and y such that x^2 ≡ a (mod pᵢ) and y^2 ≡ a (mod pᵢ).
Finding solutions for these congruences involves considering the quadratic residues modulo pᵢ. Quadratic residues are the remainders obtained when perfect squares are divided by pᵢ.
To find solutions for x^2 ≡ a (mod pᵢ) and y^2 ≡ a (mod pᵢ), we need to find the solutions for the quadratic residues of 'a' modulo pᵢ, denoted as Qᵢ. This can be done using modular arithmetic and checking the perfect squares modulo pᵢ.
By finding the quadratic residues Qᵢ and solving the congruences x^2 ≡ Qᵢ (mod pᵢ) and y^2 ≡ Qᵢ (mod pᵢ) for each prime factor, we can obtain the solutions for x and y.
It's worth noting that the existence of integer solutions depends on the values of 'a' and the quadratic residues modulo each prime factor. So, the specific expressions for x and y will vary depending on 'a' and its prime factors.