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The sequence of Giuga numbers begins 30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, … (sequence A007850 in the OEIS).. For example, 30 is a Giuga number since its prime factors are 2, 3 and 5, and we can verify that
Every composite number can be written as the product of two or more (not necessarily distinct) primes. [2] For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2 3 × 3 2 × 5; furthermore, this representation is unique up to the order of the factors.
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.
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) .
There are only 21853 pseudoprimes base 2 that are less than 2.5 × 10 10 (see page 1005 of [3]). This means that, for n up to 2.5 × 10 10, if 2 n −1 (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes. Some composite numbers (Carmichael numbers) have the property that a n − 1 is 1 (modulo n) for every a that ...
d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
In-between these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number n such that p n is an endomorphism on every Z n-algebra that can be generated as Z n-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
The number x = 2 is most often used in this basic primality check, and n = 341 = 11 × 31 is notable since , and n = 341 is the smallest composite number for which x = 2 is a false witness to primality.