# Mathematical Algorithms in TheAlgorithms/Java: Complete Guide to Number Theory, Linear Algebra, and Numerical Methods > Explore over 50 mathematical algorithms in TheAlgorithms/Java including number theory, linear algebra, and numerical methods. Discover essential math implementations for your Java projects. - Repository: [The Algorithms/Java](https://github.com/TheAlgorithms/Java) - Tags: tutorial - Published: 2026-03-04 --- **TheAlgorithms/Java provides a comprehensive library of over 50 mathematical algorithms spanning number theory, combinatorics, linear algebra, and numerical analysis, implemented as stateless utility classes in `src/main/java/com/thealgorithms/maths/`.** TheAlgorithms/Java is one of the most-starred open-source algorithm repositories on GitHub, offering production-ready implementations of classic and modern mathematical algorithms. Whether you need prime number sieves, fast Fourier transforms, or numerical integration methods, this repository provides dependency-free Java code optimized for Java 17 and above. This guide catalogs the mathematical algorithms available in TheAlgorithms/Java, organized by category with practical code examples drawn directly from the source. ## Architecture and Design Pattern All mathematical algorithms reside in the `com.thealgorithms.maths` package under `src/main/java/com/thealgorithms/maths/`. The repository follows a strict **stateless utility pattern**: each class is declared `final` with a private constructor, exposing algorithms via `public static` methods. This design eliminates instantiation overhead and ensures thread safety. The implementations rely exclusively on the Java standard library with zero external dependencies, and comprehensive JUnit test coverage exists in `src/test/java/com/thealgorithms/maths/`. ## Number Theory and Primality Algorithms ### Prime Generation and Sieving The repository contains multiple prime generation strategies. The [`SieveOfEratosthenes.java`](https://github.com/TheAlgorithms/Java/blob/main/SieveOfEratosthenes.java) class implements the classic O(n log log n) algorithm via `generatePrimes(int limit)`, returning a `List` of all primes up to the specified bound. For advanced use cases, [`SieveOfAtkin.java`](https://github.com/TheAlgorithms/Java/blob/main/SieveOfAtkin.java) provides an optimized alternative with superior asymptotic performance for large ranges. ### Divisibility and Factorization For divisibility operations, [`GCD.java`](https://github.com/TheAlgorithms/Java/blob/main/GCD.java) implements the Euclidean algorithm through `gcd(int a, int b)`, while [`LeastCommonMultiple.java`](https://github.com/TheAlgorithms/Java/blob/main/LeastCommonMultiple.java) calculates LCM using the mathematical relationship `lcm(a, b) = (a * b) / gcd(a, b)` via `lcm(int a, int b)`. These utilities handle edge cases including negative inputs and zero values. ### Special Number Sequences and Properties The collection includes generators for fundamental sequences: [`FibonacciLoop.java`](https://github.com/TheAlgorithms/Java/blob/main/FibonacciLoop.java) for Fibonacci numbers, [`CatalanNumbers.java`](https://github.com/TheAlgorithms/Java/blob/main/CatalanNumbers.java) for combinatorial structures via `catalan(int n)`, and [`BellNumbers.java`](https://github.com/TheAlgorithms/Java/blob/main/BellNumbers.java) for set partitions. Additionally, special number tests are available in [`Armstrong.java`](https://github.com/TheAlgorithms/Java/blob/main/Armstrong.java), [`HappyNumber.java`](https://github.com/TheAlgorithms/Java/blob/main/HappyNumber.java), [`PerfectNumber.java`](https://github.com/TheAlgorithms/Java/blob/main/PerfectNumber.java), and [`HarshadNumber.java`](https://github.com/TheAlgorithms/Java/blob/main/HarshadNumber.java) for validating numerical properties. ## Combinatorics and Counting The combinatorics suite provides tools for discrete mathematics calculations. [`BinomialCoefficient.java`](https://github.com/TheAlgorithms/Java/blob/main/BinomialCoefficient.java) computes nCk using dynamic programming based on Pascal's triangle via `binomialCoefficient(int n, int k)`, avoiding the overflow issues inherent in factorial division. [`CatalanNumbers.java`](https://github.com/TheAlgorithms/Java/blob/main/CatalanNumbers.java) and [`BellNumbers.java`](https://github.com/TheAlgorithms/Java/blob/main/BellNumbers.java) support advanced counting problems in graph theory and set theory applications. ## Linear Algebra and Numerical Analysis ### Matrix and Vector Operations For linear algebra computations, [`DeterminantOfMatrix.java`](https://github.com/TheAlgorithms/Java/blob/main/DeterminantOfMatrix.java) calculates matrix determinants recursively with O(n!) complexity, suitable for small to medium matrices. Vector mathematics are supported by [`VectorCrossProduct.java`](https://github.com/TheAlgorithms/Java/blob/main/VectorCrossProduct.java) for three-dimensional geometric calculations. ### Fast Fourier Transform Signal processing implementations include [`FFT.java`](https://github.com/TheAlgorithms/Java/blob/main/FFT.java), which features an iterative radix-2 Cooley-Tukey Fast Fourier Transform. The method signature `fft(ArrayList signal, boolean inverse)` requires input sizes that are powers of two and returns complex number results for frequency domain analysis. ### Numerical Integration and Root Finding Calculus utilities include [`SimpsonIntegration.java`](https://github.com/TheAlgorithms/Java/blob/main/SimpsonIntegration.java), which applies Simpson's rule for numerical integration via `integrate(DoubleUnaryOperator f, double a, double b, int n)`. Root-finding algorithms are implemented in [`SquareRootWithNewtonRaphsonMethod.java`](https://github.com/TheAlgorithms/Java/blob/main/SquareRootWithNewtonRaphsonMethod.java) and [`SquareRootWithBabylonianMethod.java`](https://github.com/TheAlgorithms/Java/blob/main/SquareRootWithBabylonianMethod.java), offering iterative approaches to approximation with configurable precision. ## Arithmetic and Exponentiation High-performance arithmetic operations are centered in [`BinaryPow.java`](https://github.com/TheAlgorithms/Java/blob/main/BinaryPow.java) and [`FastExponentiation.java`](https://github.com/TheAlgorithms/Java/blob/main/FastExponentiation.java). The `BinaryPow.binPow(int a, int p)` method implements binary exponentiation with O(log p) complexity, efficiently handling large powers without overflow through iterative bit manipulation. [`FastExponentiation.java`](https://github.com/TheAlgorithms/Java/blob/main/FastExponentiation.java) provides additional support for `long` and modular arithmetic operations. ## Statistical and Geometric Utilities ### Statistical Tools The statistics module in [`Means.java`](https://github.com/TheAlgorithms/Java/blob/main/Means.java) implements arithmetic, geometric, harmonic, and quadratic means, while [`StandardDeviation.java`](https://github.com/TheAlgorithms/Java/blob/main/StandardDeviation.java) and variance calculations support descriptive statistics. [`ZScore.java`](https://github.com/TheAlgorithms/Java/blob/main/ZScore.java) provides standard score normalization for data analysis workflows. ### Geometric Calculations Geometric algorithms include [`Area.java`](https://github.com/TheAlgorithms/Java/blob/main/Area.java) for polygon and circle area calculations, [`PythagoreanTriple.java`](https://github.com/TheAlgorithms/Java/blob/main/PythagoreanTriple.java) for generating integer triples satisfying a² + b² = c², and [`VectorCrossProduct.java`](https://github.com/TheAlgorithms/Java/blob/main/VectorCrossProduct.java) for spatial computing applications. ## Practical Implementation Examples ### Binary Exponentiation ```java import com.thealgorithms.maths.BinaryPow; public class DemoBinaryPow { public static void main(String[] args) { int base = 5; int exponent = 13; int result = BinaryPow.binPow(base, exponent); System.out.println(base + "^" + exponent + " = " + result); // Output: 5^13 = 1220703125 } } ``` ### Greatest Common Divisor ```java import com.thealgorithms.maths.GCD; public class DemoGCD { public static void main(String[] args) { System.out.println("GCD(48, 180) = " + GCD.gcd(48, 180)); // Output: GCD(48, 180) = 12 } } ``` ### Prime Generation with Sieve of Eratosthenes ```java import com.thealgorithms.maths.SieveOfEratosthenes; import java.util.List; public class DemoSieve { public static void main(String[] args) { List primes = SieveOfEratosthenes.generatePrimes(50); System.out.println(primes); // Output: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] } } ``` ### Fast Fourier Transform ```java import com.thealgorithms.maths.FFT; import com.thealgorithms.maths.FFT.Complex; import java.util.ArrayList; import java.util.List; public class DemoFFT { public static void main(String[] args) { // Example signal: 4 samples (must be power of two) List signal = List.of( new Complex(0, 0), new Complex(1, 0), new Complex(0, 0), new Complex(-1, 0)); ArrayList transformed = FFT.fft(new ArrayList<>(signal), false); System.out.println("FFT result:"); transformed.forEach(c -> System.out.println(c.getReal() + " + " + c.getImaginary() + "i")); } } ``` ### Numerical Integration with Simpson's Rule ```java import com.thealgorithms.maths.SimpsonIntegration; public class DemoSimpson { public static void main(String[] args) { // Integrate f(x) = x^2 from 0 to 1 double integral = SimpsonIntegration.integrate(x -> x * x, 0, 1, 100); System.out.println("∫₀¹ x² dx ≈ " + integral); // Expected value: 1/3 ≈ 0.33333 } } ``` ## Summary - **TheAlgorithms/Java** organizes mathematical algorithms in `src/main/java/com/thealgorithms/maths/` within the `com.thealgorithms.maths` package. - All implementations follow a **stateless utility pattern** with `public static` methods, final classes, and private constructors to prevent instantiation. - The repository covers **eight major categories**: number theory (primes, GCD, special numbers), combinatorics (binomial coefficients, Catalan numbers), linear algebra (determinants, FFT), numerical methods (Simpson's integration, root finding), exponentiation (binary pow), arithmetic, statistics, and geometry. - **Zero dependencies** are required beyond the Java standard library, with full compatibility for Java 17+. - Each algorithm includes corresponding **JUnit tests** in `src/test/java/com/thealgorithms/maths/` to validate edge cases and correctness. ## Frequently Asked Questions ### How do I import and use mathematical algorithms from TheAlgorithms/Java in my project? Import the specific utility class from the `com.thealgorithms.maths` package and invoke the static method directly. For example, use `import com.thealgorithms.maths.GCD;` followed by `int result = GCD.gcd(48, 180);`. No instantiation or dependency configuration is required, as all algorithms are implemented as stateless utility functions. ### What Java version is required to run these mathematical algorithms? TheAlgorithms/Java targets **Java 17 and above**. The mathematical algorithms in `src/main/java/com/thealgorithms/maths/` utilize modern Java features such as `DoubleUnaryOperator` for numerical integration and enhanced `List` interfaces, ensuring compatibility with current Long-Term Support (LTS) versions. ### Are the mathematical algorithms in TheAlgorithms/Java thread-safe? Yes, all mathematical utility classes are **inherently thread-safe**. Because each class is stateless with only static methods and no instance variables, concurrent threads can safely invoke `BinaryPow.binPow()`, `FFT.fft()`, or `SimpsonIntegration.integrate()` without synchronization concerns or race conditions. ### How can I contribute a new mathematical algorithm to the repository? Create a new class in `src/main/java/com/thealgorithms/maths/` following the existing **stateless pattern**: declare the class `final`, add a private constructor to prevent instantiation, and implement the algorithm as a `public static` method. Include comprehensive JUnit tests in `src/test/java/com/thealgorithms/maths/` covering edge cases such as zero inputs, negative values, and maximum integer bounds.