Prime Checker 5.0

This application provides a comprehensive set of mathematical calculations for a given integer, focusing on various important number-theoretic functions. The features it supports include:

• Integer factorization
• Euler's totient function
• Identification of primitive roots, if they exist
• Calculation of x and y such that x² + y² = p when p conforms to the form 4n + 1
• Calculation of x and y such that x² + 3y² = p when p conforms to the form 3n + 1
• Calculation of x and y such that x² + 2y² = p when p conforms to the forms 8n + 1 or 8n + 3
• Calculation of x and y such that x² - 2y² = p when p conforms to the forms 8n + 1 or 8n + 7

To achieve these computations, the application employs several sophisticated algorithms, including Fermat's algorithm, Pollard's rho algorithm, and Lenstra's ECM for factorization, as well as the Miller–Rabin algorithm for primality testing. This integration of methodologies enhances its reliability and efficacy in performing complex mathematical calculations.