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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).
Some composite numbers (Carmichael numbers) have the property that a n − 1 is 1 (modulo n) for every a that is coprime to n. The smallest example is n = 561 = 3·11·17, for which a 560 is 1 (modulo 561) for all a coprime to 561.
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.
[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 ...
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
Note: Numbers in bold are themselves highly composite numbers. Only the twentieth highly composite number 7560 (= 3 × 2520) is absent. 10080 is a so-called 7-smooth number (sequence A002473 in the OEIS).
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The quadratic residuosity problem (QRP [1]) in computational number theory is to decide, given integers and , whether is a quadratic residue modulo or not. Here N = p 1 p 2 {\displaystyle N=p_{1}p_{2}} for two unknown primes p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} , and a {\displaystyle a} is among the numbers which are not ...