quadratic residue
C2Technical / Academic
Definition
Meaning
In number theory, an integer q is called a quadratic residue modulo n if there exists an integer x such that x² ≡ q (mod n).
A number which gives a perfect square when considered modulo a given integer, i.e., a number that is congruent to a perfect square modulo n. The concept is fundamental in number theory and has applications in cryptography and computational mathematics.
Linguistics
Semantic Notes
Always a compound noun. The concept is exclusive to the field of number theory and advanced algebra. Requires understanding of modular arithmetic.
Dialectal Variation
British vs American Usage
Differences
No lexical differences. Potential minor pronunciation differences in 'quadratic' and 'residue'.
Connotations
Identical technical connotations.
Frequency
Used with identical frequency and context in academic mathematics globally. No regional preference.
Vocabulary
Collocations
Grammar
Valency Patterns
[Number] is a quadratic residue modulo [number].To determine the quadratic residues of [modulus].The quadratic residue symbol (a/p).Vocabulary
Synonyms
Neutral
Vocabulary
Antonyms
Usage
Context Usage
Business
Not used.
Academic
Exclusively used in mathematics papers, textbooks, and advanced courses in number theory and cryptography.
Everyday
Not used.
Technical
Core term in cryptographic algorithms (e.g., Goldwasser-Micali cryptosystem) and computational number theory.
Examples
By Part of Speech
adjective
British English
- The quadratic-residue properties were analysed.
- A quadratic-residue sequence was constructed.
American English
- The quadratic-residue properties were analyzed.
- A quadratic-residue code was implemented.
Examples
By CEFR Level
- In modular arithmetic, 4 is a quadratic residue modulo 7 because 2² = 4.
- The cryptosystem's security relies on the computational difficulty of distinguishing quadratic residues from non-residues modulo a composite number.
- Euler's criterion provides a direct method for determining whether an integer is a quadratic residue modulo an odd prime.
Learning
Memory Aids
Mnemonic
Think: 'Quad-RAT-ic Re-SID-ue' – A RAT (rodent) might leave a residue, and a QUAD (square) RAT is a square number. The residue (what's left) when you consider squares modulo n.
Conceptual Metaphor
A 'residue' is a leftover or remainder. A 'quadratic residue' is a remainder that is a perfect square's leftover after division by a modulus.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Прямой перевод «квадратичный вычет» является точным и стандартным. Не переводите как «квадратный остаток».
Common Mistakes
- Using 'quadratic residual' (incorrect noun form).
- Confusing with 'prime residue' or other residue classes.
- Omitting the crucial 'modulo [n]' specification.
Practice
Quiz
What is a quadratic residue modulo 11?
FAQ
Frequently Asked Questions
No, because there is no integer x such that x² ≡ 2 (mod 7). The quadratic residues modulo 7 are 1, 2, and 4.
A quadratic residue is congruent to a square modulo n. A quadratic non-residue is any integer modulo n that is not congruent to a square.
Problems like the Quadratic Residuosity Problem are believed to be computationally hard, forming the basis for provably secure cryptographic systems such as the Goldwasser-Micali cryptosystem.
A foundational theorem in number theory that establishes a relationship between the solvability of the congruences x² ≡ p (mod q) and x² ≡ q (mod p) for distinct odd primes p and q.