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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.
For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively. A related concept is that of a largely composite number , a positive integer that has at least as many divisors as all smaller positive integers.
4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... (sequence A059754 in the OEIS). Smith numbers can be constructed from factored repunits. [5] [verification needed] As of 2010, the largest known Smith number in base 10 is 9 × R 1031 × (10 4594 + 3 × 10 2297 + 1) 1476 × 10 3 913 210. where R 1031 is the base 10 repunit (10 1031 ...
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) .
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.
The next number not yet crossed out in the list after 5 is 7; the next step would be to cross out every 7th number in the list after 7, but they are all already crossed out at this point, as these numbers (14, 21, 28) are also multiples of smaller primes because 7 × 7 is greater than 30.
In number theory, a superior highly composite number is a natural number which, ... 2 4 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 : 4,2,1,1,1 120 2 2 ⋅ 6 ⋅ 2310 : 10 720720:
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.