Understanding the Equation x^2 - y^2 2000^2 and Its Integral Solutions

In the realm of number theory, the equation x^2 - y^2 2000^2 presents an intriguing challenge to find all possible integral solutions. This article delves into the methodology and explores the process step-by-step to uncover the total number of integral pairs (x, y) that satisfy this equation.

Factoring the Left-Hand Side

The approach to solving the equation x^2 - y^2 2000^2 begins by factoring the left-hand side of the equation:

[x^2 - y^2 (x y)(x - y) 2000^2]

Defining a x y and b x - y, we rewrite the equation as:

[ab 2000^2]

Identifying Divisors of 4000000

The next step involves finding pairs of integers (a, b) whose product is 2000^2 4000000. The prime factorization of 2000 is:

[2000 2^4 times 5^3]

Thus, the prime factorization of 2000^2 is:

[2000^2 2^8 times 5^6]

The formula for the number of divisors is:

[text{Number of divisors} (8 1) times (6 1) 9 times 7 63]

Each pair (a, b) corresponds to a unique solution for x and y via:

[x frac{a b}{2} quad y frac{b - a}{2}]

Ensuring Both x and y are Integers

For both x and y to be integers, a b and b - a must both be even. Since ab 4000000 is even, a and b must both be even. Consequently, a 2m and b 2n, which leads to:

[2m cdot 2n 4000000 implies 4mn 4000000 implies mn 1000000]

Prime Factorization and Divisors of 1000000

The prime factorization of 1000000 is:

[1000000 10^6 2^6 times 5^6]

The number of divisors of 1000000 is:

[6 times 7 42]

Each divisor pair (m, n) gives a corresponding (a, b) via a 2m and b 2n. Each pair leads to a unique solution for x and y. Since both positive and negative divisors yield the same solutions, we have:

[42 times 2 84]

However, we must account for the sign of a and b. Hence, the total number of integral solutions is:

[84 times 2 168]

Final Total Number of Integral Solutions

The total number of integral solutions of the equation x^2 - y^2 2000^2 is:

[boxed{168}]

Understanding and solving such equations provides valuable insights into number theory and strengthens problem-solving skills. The exploration of integral solutions, particularly in the context of x^2 - y^2 2000^2, showcases the beauty and complexity of mathematical equations.