Advanced Math Tools
A collection of advanced mathematical calculators and utilities for number theory, combinatorics, sequences, and more.
Number Theory
Tools for exploring prime numbers, divisibility, modular arithmetic, and the fundamental properties of integers.
Check if a number is prime with an efficient algorithm.
Decompose any positive integer into its prime factors with step-by-step explanation.
Calculate φ(n) - the count of integers from 1 to n that are coprime to n.
Calculate τ(n) and σ(n) - the count and sum of all divisors of a positive integer.
Find the GCD and Bézout coefficients of two numbers.
Calculate a^b mod m efficiently using binary exponentiation.
Solve systems of linear congruences with step-by-step solutions.
Calculate μ(n) - a fundamental function in number theory used in the Mobius inversion formula.
Express any proper fraction as a sum of distinct unit fractions using the greedy algorithm.
Calculate the Jacobi symbol (a/n), a generalization of the Legendre symbol used in primality testing.
Check if a number is perfect, abundant, or deficient by analyzing the sum of its proper divisors.
Find primitive roots modulo n, fundamental to number theory and cryptographic protocols like Diffie-Hellman.
Calculate the digital root (repeated digit sum) of any number, related to divisibility by 9 and casting out nines.
Check if a number is happy by repeatedly summing the squares of its digits until reaching 1 or entering a cycle.
Check if a number equals the sum of its digits each raised to the power of the digit count (Armstrong number).
Calculate lambda(n) - the smallest positive integer m such that a^m = 1 (mod n) for all a coprime to n.
Express any even number as the sum of two prime numbers and find all possible prime pairs.
Check if a number is a Kaprekar number - where the square can be split into two parts that sum to the original.
Check if a number is a Harshad (Niven) number - divisible by the sum of its digits.
Check if a number is a Smith number - where the digit sum equals the digit sum of its prime factorization.
Calculate ord_n(a) - the smallest positive integer k such that a^k ≡ 1 (mod n), fundamental in cryptography.
Check if a number is a palindrome (reads the same forwards and backwards) in different bases like binary, decimal, or hexadecimal.
Combinatorics
Counting, arrangements, partitions, and the mathematics of discrete structures.
Calculate nPr and nCr with large number support.
Calculate D(n) - the number of permutations where no element appears in its original position.
Expand (a + b)^n using the binomial theorem with step-by-step coefficients.
Calculate the number of ways to partition a positive integer into sums of positive integers.
Generate and count Dyck paths for combinatorial analysis.
Calculate r₅(n) - the number of ways to represent n as a sum of five squares.
Calculate Stirling numbers of the first and second kind for counting permutations with cycles and set partitions.
Calculate Catalan numbers that count balanced parentheses, binary trees, and many other combinatorial structures.
Calculate Bell numbers that count the number of ways to partition a set into non-empty subsets.
Sequences
Explore famous sequences, iterative processes, and convergent series.
Calculate Fibonacci numbers and explore the golden ratio.
Calculate Lucas numbers and explore their connection to the golden ratio and the Lucas-Lehmer primality test.
Explore the famous 3n+1 problem by generating Collatz sequences and analyzing their behavior.
Convert numbers to continued fraction representation and compute convergents.
Algebra
Solve equations, work with polynomials, and convert between number systems.
Geometry
Calculate areas, visualize curves, and work with geometric shapes.
Linear Algebra
Matrix operations and transformations for vectors and linear systems.
Numerical Methods
Approximation techniques, simulations, and computational approaches to mathematical problems.