Legendre Symbol and Reciprocity
Test residues and relate primes.
Key idea
Test residues and relate primes.
In depth
(a/p) = 1, −1 or 0 depending on whether a is a QR, a non-residue, or divisible by p. Quadratic reciprocity: for odd primes p, q, (p/q)(q/p) = (−1)^((p−1)(q−1)/4). It allows quick computation.
Worked example
(3/7) = −(7/3) = −(1/3) = −1, so 3 is not a QR mod 7.
Real-world uses
- Quickly testing whether a number is a square modulo a prime in algorithms
Common mistake
Forgetting the sign factor when both primes are 3 mod 4.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Legendre Symbol and Reciprocity?
Test residues and relate primes. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Legendre Symbol and Reciprocity used in real life?
Real-world examples: Quickly testing whether a number is a square modulo a prime in algorithms.
What should I learn before Legendre Symbol and Reciprocity?
Make sure you are comfortable with Quadratic Residues, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Legendre Symbol and Reciprocity?
Forgetting the sign factor when both primes are 3 mod 4.
How can I practise Legendre Symbol and Reciprocity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Legendre Symbol and Reciprocity?
Move on to Primitive Roots next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.