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[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 ...
The JScience library has a Complex number class. The JAS library allows the use of complex numbers. Netlib has a complex number class for Java. javafastcomplex also adds complex number support for Java; jcomplexnumber is a project on implementation of complex number in Java. JLinAlg includes complex numbers with arbitrary precision.
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".
An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.. Theorem (A. Korselt 1899): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of , it is true that .
The sequence of 1-Brazilian numbers is composed of other primes, the only square of prime that is Brazilian, 121, and composite numbers ≥ 8 that are the product of only two distinct factors such that n = a × b = aa b–1 with 1 < a < b – 1. (sequence A288783 in the OEIS).
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.
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In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free , the factorization is written without exponents, writing the repeated factor as many times as needed.