Search results
Results From The WOW.Com Content Network
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4) .
The Miller–Rabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (Miller–Rabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.
No composite number below 2 64 (approximately 1.845·10 19) passes the strong or standard Baillie–PSW test, [3] that result was also separately verified by Charles Greathouse in June 2011. Consequently, this test is a deterministic primality test on numbers below that bound.
For other numbers, the algorithm only returns “composite” with no further information. For example, consider n = 341 and a = 2. We have n − 1 = 85 × 4. Then 2 85 mod 341 = 32 and 32 2 mod 341 = 1. This tells us that n is a pseudoprime base 2, but not a strong pseudoprime base 2.
Inputs: n: a value to test for primality, n>3; k: a parameter that determines the number of times to test for primality Output: composite if n is composite, otherwise probably prime Repeat k times: Pick a randomly in the range [2, n − 2] If (), then return composite
[1] [2] Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. [3] [4] E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself ...
Check if n is a perfect power: if n = a b for integers a > 1 and b > 1, then output composite. Find the smallest r such that ord r (n) > (log 2 n) 2. If r and n are not coprime, then output composite. For all 2 ≤ a ≤ min (r, n−1), check that a does not divide n: If a|n for some 2 ≤ a ≤ min (r, n−1), then output composite.
So if it is unknown whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol ( a / n ) and compare it with Euler's formula; if they differ modulo n, then n is composite; if they have the same residue modulo n for many different values of a, then n is "probably prime".