Quadratic Residue Calculator
Determine quadratic residues and calculate Legendre symbols for number theory analysis
About this calculator
A quadratic residue calculator determines whether an integer is a quadratic residue modulo a prime number and computes Legendre symbols for number theory analysis. This tool is essential for students and researchers studying modular arithmetic, cryptography, and advanced mathematics. It simplifies complex calculations involving quadratic congruences and helps verify theoretical results in algebraic number theory, making it invaluable for academic research and mathematical problem-solving.
How to use
Enter the integer you want to test and the prime modulus into the input fields. The calculator will automatically determine if the integer is a quadratic residue modulo the prime and display the corresponding Legendre symbol value (1, -1, or 0) along with detailed explanations of the calculation process.
Frequently asked questions
What is a quadratic residue?
A quadratic residue is an integer that has a square root modulo a prime number. If x² ≡ a (mod p) has a solution, then a is a quadratic residue modulo p.
What does the Legendre symbol represent?
The Legendre symbol (a/p) equals 1 if a is a quadratic residue mod p, -1 if not, and 0 if a ≡ 0 (mod p).
Why are quadratic residues important in mathematics?
Quadratic residues are fundamental in number theory, cryptography, and coding theory. They help solve quadratic congruences and are used in primality testing algorithms.