The given equation is of the form:

\[
x^2 - N y^2 = 1
\]

This is a **Pell's equation**, where \( N = 114514 \). Pell's equation is a type of Diophantine equation, and its general solution involves finding the integer values of \( x \) and \( y \) that satisfy the equation.

### General Solution Approach:

1. **Find the fundamental solution**: Pell's equation often has a smallest solution \( (x_1, y_1) \), called the fundamental solution. Once you have the fundamental solution, you can generate infinitely many solutions.

2. **Use continued fractions**: The solutions to Pell's equation are connected to the continued fraction expansion of \( \sqrt{N} \). The smallest solution is often obtained from this expansion.

3. **Generate further solutions**: Once you have the fundamental solution, all other solutions can be derived using the recursive relation:
   \[
   x_{n+1} = x_1 \cdot x_n + N \cdot y_1 \cdot y_n
   \]
   \[
   y_{n+1} = x_1 \cdot y_n + y_1 \cdot x_n
   \]

Would you like to proceed with finding the fundamental solution for this specific equation, or would you like more details on any particular step?

### Related Questions:
1. How do continued fractions relate to Pell's equation?
2. What are Diophantine equations, and how are they solved?
3. Can Pell's equation always be solved for any integer \( N \)?
4. What is the history behind Pell's equation?
5. How does the recursive formula for solutions work in Pell's equation?

**Tip:** Pell's equations have infinitely many solutions, but finding the fundamental solution is crucial, as all others are built from it.

Explore how to solve the Pell's equation x^2 - 114514*y^2 = 1 using continued fractions and recursive formulas. Learn about Diophantine equations, their solutions, and the use of fundamental solutions. Suitable for students in undergraduate math courses.

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