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Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is ...
Output prime. Here ord r (n) is the multiplicative order of n modulo r, log 2 is the binary logarithm, and () is Euler's totient function of r. Step 3 is shown in the paper as checking 1 < gcd(a,n) < n for all a ≤ r. It can be seen this is equivalent to trial division up to r, which can be done very efficiently without using gcd.
The idea here is to find an m that is divisible by a large prime number q. This prime is a few digits smaller than m (or N) so q will be easier to prove prime than N. Assuming we find a curve which passes the criterion, proceed to calculate mP and kP. If any of the two calculations produce an undefined expression, we can get a non-trivial ...
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
Input #1: b, the number of bits of the result Input #2: k, the number of rounds of testing to perform Output: a strong probable prime n while True: pick a random odd integer n in the range [2 b −1 , 2 b −1] if the Miller–Rabin test with inputs n and k returns “ probably prime ” then return n
A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially, let p equal 2, the smallest prime number.
BASIC extensions See also References External links Dialects 0–9 1771-DB BASIC Allen-Bradley PLC industrial controller BASIC module; Intel BASIC-52 extended with PLC-specific calls. 64K BASIC Cross-platform, interactive, open-source interpreter for microcomputer BASIC. A ABasiC (Amiga) Relatively limited. Initially provided with Amiga computers by MetaComCo. ABC BASIC designed for the ABC 80 ...
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...