Step by Step solution about How to resolve the algorithm Pell's equation step by step in the Mathematica/Wolfram Language programming language

Source code Explanation:

This Wolfram code uses the FindInstance function to find positive integer solutions to Diophantine equations of the form:

x^2 - k * y^2 = 1

where k is a constant.

Specific Instances:

The code finds solutions for four different values of k:

1. k = 61
2. k = 109
3. k = 181
4. k = 277

Output:

For each value of k, the code outputs the following information:

FindInstance[x^2 - k * y^2 == 1, {x, y}, PositiveIntegers]

This line calls the FindInstance function to find solutions to the equation x^2 - k * y^2 == 1 within the set of positive integers.

Example Output:

For the value k = 61, the output might look like this:

{{x -> 17}, {x -> 569}, {x -> 7441}, ...}

This output shows that the equation x^2 - 61 * y^2 == 1 has multiple positive integer solutions, including x = 17, x = 569, and x = 7441.

FindInstance[x^2 - 61 y^2 == 1, {x, y}, PositiveIntegers]
FindInstance[x^2 - 109 y^2 == 1, {x, y}, PositiveIntegers]
FindInstance[x^2 - 181 y^2 == 1, {x, y}, PositiveIntegers]
FindInstance[x^2 - 277 y^2 == 1, {x, y}, PositiveIntegers]