Calculating Legendre Symbols

Number Theory · Axiom Academy

EXAMPLE Calculating Legendre Symbols Master the computation of Legendre symbols using Euler's criterion and key properties Excellent work! You've mastered calculating Legendre symbols. Here's what we learned: Euler's Criterion: The Legendre symbol (a/p) ≡ a^((p-1)/2) (mod p) provides a computational method to determine quadratic residuosity. Efficient Computation: Successive squaring (repeated squaring method) allows us to compute large powers modulo p efficiently without calculating the full power. Interpretation: If the result is 1, then a is a quadratic residue mod p. If it's -1 (or p-1), then a is a non-residue. This tells us whether x² ≡ a (mod p) has solutions. Multiplicativity: The Legendre symbol is multiplicative, meaning (ab/p) = (a/p)(b/p), which allows us to factor and simplify complex symbols. Alternative Methods: For larger primes, quadratic reciprocity and other properties can provide faster methods than direct computation via Euler's criterion. Practice computing Legendre symbols with different values to build confidence! Try using both Euler's criterion and properties like quadratic reciprocity for comparison.

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