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To test divisibility by any number expressed as the product of prime factors , we can separately test for divisibility by each prime to its appropriate power. For example, testing divisibility by 24 (24 = 8 × 3 = 2 3 × 3) is equivalent to testing divisibility by 8 (2 3 ) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 ...
Fermat primality test. Pseudoprime; Carmichael number; Euler pseudoprime; Euler–Jacobi pseudoprime; Fibonacci pseudoprime; Probable prime; Baillie–PSW primality test; Miller–Rabin primality test; Lucas–Lehmer primality test; Lucas–Lehmer test for Mersenne numbers; AKS primality test
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.
Two properties of 1001 are the basis of a divisibility test for 7, 11 and 13. The method is along the same lines as the divisibility rule for 11 using the property 10 ≡ -1 (mod 11). The two properties of 1001 are 1001 = 7 × 11 × 13 in prime factors 10 3 ≡ -1 (mod 1001) The method simultaneously tests for divisibility by any of the factors ...
Tests that show whether a large number can be divided by a number from 2 to 12 1962 Oct: A collection of puzzles involving numbers, logic, and probability 1962 Nov: Some puzzles based on checkerboards: 1962 Dec: Some simple tricks and manipulations from the ancient lore of string play: 1963 Jan: The author pays his annual visit to Dr. Matrix ...
If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1 ⁄ 4, in which case p is a strong pseudoprime, and a is a strong liar. Therefore after k non-conclusive random tests, the probability that p is composite is at most 4 −k, and may thus be made as low as desired by increasing k.
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