Pell's Equations

Number-Theory

Definition

Pell’s equations are equations in the form where is not a square. There are infinite integer solutions to the equation.

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Norm

Define a number , then the conjugate of is . The conjugate is multiplicative. The Norm of is given as . Note the resemblance to Complex Numbers. The Norm is multiplicative - . This leads us to seeing that is the same as Using the fact that is multiplicative, . So from one solution we can generate infinitely many.

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Fundamental Solution

We can find a fundamental solution such that every solution is found by

Proof

1. Consider to be the smallest real in the for and .
2. Consider such that is another solution to .
3. Let be such that . This is guaranteed to exist as is continuous and these inequalities cover the whole domain.
4. Since
5. Define . Since is multiplicative .
6. Expanding gets something in the for . Since , .
7. Hence .
8. We can also prove that as and by AM-GM . ( by the same algebra in step 4 since ).
9. Since the minimality of is contradicted unless which only happens when as desired.

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General and Recursive Solutions of Pell’s Equations

INSERT PROOF Let be the fundamental solution and be the th solution, then

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Negative Pell’s Equations