Miller Rabin Primality Test Calculator
Test if large numbers are prime using the Miller-Rabin probabilistic primality test. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateWitness Test Results
Formula
The Miller-Rabin test decomposes n-1 as 2^r * d (d odd), then for each test base a, checks if a^d mod n = 1 or if any a^(2^i * d) mod n = n-1. If neither holds, n is composite. Each test reduces the error probability by a factor of 4.
Last reviewed: December 2025
Worked Examples
Example 1: Testing 104729 for Primality
Example 2: Testing 561 (Carmichael Number)
Background & Theory
The Miller-Rabin Primality Test Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Miller-Rabin Primality Test Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
Test: a^d mod n and a^(2^i * d) mod n for i = 0 to r-1
The Miller-Rabin test decomposes n-1 as 2^r * d (d odd), then for each test base a, checks if a^d mod n = 1 or if any a^(2^i * d) mod n = n-1. If neither holds, n is composite. Each test reduces the error probability by a factor of 4.
Worked Examples
Example 1: Testing 104729 for Primality
Problem: Use the Miller-Rabin test with 10 iterations to determine if 104729 is prime.
Solution: n = 104729, n-1 = 104728\nDecompose: 104728 = 2^3 x 13091, so r = 3, d = 13091\nTest base 2: 2^13091 mod 104729 = check sequence\nTest base 3: 3^13091 mod 104729 = check sequence\n... (continue for remaining bases)\nAll bases pass the primality conditions.\nError bound: (1/4)^10 = 9.5367e-7\nConfidence: 99.9999%
Result: Probably Prime | Confidence: 99.9999% | 10 iterations
Example 2: Testing 561 (Carmichael Number)
Problem: Test 561, the smallest Carmichael number, using Miller-Rabin.
Solution: n = 561, n-1 = 560\nDecompose: 560 = 2^4 x 35, so r = 4, d = 35\nTrial division: 561 / 3 = 187 (factor found!)\n561 = 3 x 11 x 17\nNote: 561 passes the Fermat test for all coprime bases\nbut Miller-Rabin detects it as composite.\nFermat test alone would incorrectly identify it as prime.
Result: Composite | Factor found: 3 | 561 = 3 x 187
Frequently Asked Questions
How reliable is the Miller-Rabin test and what is the probability of error?
The Miller-Rabin test has a worst-case error probability of at most 1/4 per iteration, meaning each additional random base tested reduces the probability of incorrectly declaring a composite number as prime by a factor of 4. With k iterations, the error probability is at most (1/4)^k. For 10 iterations, the error probability is less than 1 in a million. For 20 iterations, it drops below 1 in a trillion. In practice, the actual error rate is much lower than this theoretical bound because most composite numbers have many witnesses. The test never produces false negatives: if it says a number is composite, that determination is always correct. Only the probably prime result carries uncertainty.
What is the difference between deterministic and probabilistic primality testing?
Deterministic primality tests, such as the AKS algorithm discovered in 2002, produce a guaranteed correct answer for any input with zero probability of error. However, deterministic tests are generally slower for very large numbers. Probabilistic tests like Miller-Rabin trade absolute certainty for dramatically faster execution. The Miller-Rabin test runs in O(k * log^2(n)) time, making it practical for numbers with hundreds of digits. For numbers below certain thresholds, specific sets of bases make Miller-Rabin deterministic: testing bases 2, 3, 5, and 7 gives correct results for all numbers below 3,215,031,751. The practical reality is that probabilistic tests with sufficient iterations are reliable enough for cryptographic applications where they are used to generate RSA key pairs.
Why is primality testing important for cryptography and computer security?
Modern public-key cryptography, particularly the RSA algorithm, relies fundamentally on the difficulty of factoring large numbers that are the product of two large primes. Generating RSA keys requires finding two large prime numbers, typically 512 to 2048 bits each. The Miller-Rabin test is the industry-standard method for this because it can quickly verify primality of numbers with hundreds of digits. If primality testing were unreliable, cryptographic keys could be generated from non-prime numbers, making them trivially breakable. The security of internet banking, encrypted communications, and digital signatures all depend on fast, reliable primality testing. Most cryptographic libraries use 40 or more iterations of Miller-Rabin, making the error probability astronomically small.
What are witnesses and liars in the context of Miller-Rabin testing?
In Miller-Rabin terminology, a witness is a base value a that proves a number is composite. If testing base a against number n shows that n fails the primality conditions, then a is called a witness to the compositeness of n. A liar is a base that makes a composite number appear to pass the primality test. For any composite odd number n, at most 1/4 of all possible bases between 2 and n-2 are liars, which is why the error probability per iteration is at most 25 percent. Strong pseudoprimes are composite numbers that have many liars, making them harder to detect as composite. The smallest strong pseudoprime to base 2 is 2047, and it requires testing with base 3 as well to correctly identify it as composite.
How does the decomposition of n-1 into 2^r * d work in the Miller-Rabin test?
The decomposition of n-1 into the form 2^r * d where d is odd is a crucial first step of the Miller-Rabin algorithm. Since n is odd and greater than 2, n-1 is even and can be repeatedly divided by 2. The exponent r counts how many times n-1 is divisible by 2, and d is the remaining odd factor. For example, if n = 221, then n-1 = 220 = 4 * 55 = 2^2 * 55, giving r = 2 and d = 55. This decomposition allows the algorithm to check a sequence of modular exponentiations: a^d, a^(2d), a^(4d), up to a^(2^(r-1) * d). These values form a chain where each is the square of the previous one modulo n, and the pattern of these values reveals whether n is prime or composite.
What is the relationship between Fermat's Little Theorem and the Miller-Rabin test?
The Miller-Rabin test is a strengthened version of the Fermat primality test, which is based on Fermat's Little Theorem stating that if p is prime and a is not divisible by p, then a^(p-1) mod p = 1. The Fermat test simply checks whether a^(n-1) mod n equals 1, but it fails for Carmichael numbers, which are composite numbers that pass the Fermat test for all bases coprime to them. The smallest Carmichael number is 561. Miller-Rabin strengthens this by examining the intermediate values during the computation of a^(n-1) mod n, specifically looking at the square root chain derived from the 2^r * d decomposition. This additional analysis catches Carmichael numbers and dramatically reduces false positives compared to the basic Fermat test.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy