Overview
Sieve of Atkin is a number theory method for prime generation. It works by using modular quadratic forms to mark candidate primes.
ExtendedSource checked
Sieve of Atkin
Sieve of Atkin addresses prime generation by using modular quadratic forms to mark candidate primes.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Intermediate
- Family
- Number Theory
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 0
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
prime generationnumber theorymodular arithmeticprimality testingnumerical methods
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n log n)
Worst
O(n^2)
Space
O(n)
When to Use
- Use Sieve of Atkin when solving prime generation and you want to benefit from using modular quadratic forms to mark candidate primes.
- Choose it when O(n log n) average/worst-case time and O(n) memory match your workload profile.
- Prefer it over Extended Euclidean Algorithm or Binary GCD when the problem structure favors prime generation.
When Not to Use
- Avoid when the input is ill-conditioned and the method lacks a stability safeguard.
- Avoid when exact symbolic results are required instead of numeric approximations.
What It Does
- Computes prime generation within Number Theory workloads (O(n log n) time, O(n) space).
- Works by using modular quadratic forms to mark candidate primes.
- Maintains key invariants across prime generation, number theory, modular arithmetic.
Use Cases
- Production and research workloads requiring prime generation via using modular quadratic forms to mark candidate primes.
- Number Theory pipelines where O(n log n) time and O(n) space fit system constraints.
- Engineering comparisons against Extended Euclidean Algorithm and Binary GCD when tuning prime generation.
Inputs
- Numbers, vectors, matrices, or functions
- Tolerance or precision settings
Outputs
- Computed numeric result
- Residual, factorization, or approximation
Pseudocode
- 1.Validate input dimensions, conditioning, and numerical tolerance parameters for Sieve of Atkin.
- 2.Initialize the working buffers, basis vectors, or decomposition factors (prime generation).
- 3.Execute the core numerical recurrence by using modular quadratic forms to mark candidate primes.
- 4.Monitor residual error, stability bounds, or step convergence across iterations.
- 5.Return the computed prime generation coefficients, factors, or transformed values.
Canonical Example
- Input
- Numbers, vectors, matrices, or functions configured for a representative prime generation test case
- Output
- Computed numeric result satisfying Sieve of Atkin invariants
- Explanation
- Sieve of Atkin processes the input by using modular quadratic forms to mark candidate primes, tracking prime generation and number theory to produce the verified prime generation result.
Implementation Notes
- Document parameter meanings such as tolerance, precision, and iteration limit.
- Use residual or error checks rather than trusting iteration count alone.
- Prefer stable formulations when inputs are ill-conditioned.
Common Pitfalls
- Ignoring overflow, cancellation, or loss of precision.
- Applying the method outside its stability assumptions.
- Testing only well-conditioned inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow using modular quadratic forms to mark candidate primes on a small prime generation example.
Step 1 of 5: Validate input dimensions, conditioning, and numerical tolerance parameters for Sieve of Atkin.
numerical-methods-mechanismNotes
Sieve of Atkin operates in O(n log n) time and O(n) space within the Number Theory family. Often compared with Extended Euclidean Algorithm and Binary GCD.
Sources (2)
What each reference supports for this entry.
Modern Computer Arithmetic
BookReference for Sieve of Atkin's arithmetic operations, modular arithmetic, multiplication, division, or number-theoretic implementation details.
Open sourceIntroduction to Algorithms (CLRS)
BookTextbook support for Sieve of Atkin's algorithm family, complexity framing, and standard implementation concepts.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
AlternativesSieve of Eratosthenes
Numerical Methods · Foundation
Sieve of Atkin is a more advanced prime sieve with different arithmetic filters.
Compare with thisConceptual neighborExtended Euclidean Algorithm
Numerical Methods · Foundation
Sieve of Atkin and Extended Euclidean Algorithm share a concrete number theory mechanism for prime generation.
Compare with thisRelationships
View full relationship pageConceptual neighborExtended Euclidean Algorithm
Sieve of Atkin and Extended Euclidean Algorithm share a concrete number theory mechanism for prime generation.
Conceptual neighborBinary GCD
Sieve of Atkin and Binary GCD share a concrete number theory mechanism for prime generation.
Conceptual neighborChinese Remainder Theorem Algorithm
Sieve of Atkin and Chinese Remainder Theorem Algorithm share a concrete number theory mechanism for prime generation.
Related Algorithms
Tags
Numerical Methods