A composite number

is a positive integer
1" /> i beg your pardon is not prime (i.e., which has components other 보다 1 and itself). The first few composite number (sometimes dubbed "composites" for short) space 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (OEIS A002808), whose prime decompositions space summarized in the following table. Keep in mind that the number 1 is a special situation which is thought about to be neither composite no one prime.

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 prime factorization prime factorization 4 20 6 21 8 22 9 24 10 25 12 26 14 27 15 28 16 30 18 32

The

th composite number
have the right to be created using the sdrta.net Language code

Composite := FixedPoint + 1&, n>The Dirichlet generating duty of the characteristic function of the composite numbers

is provided by

 )/(n^s)" /> (1) (2) (3)

where

is the Riemann zeta function,
is the prime zeta function, and also
" /> is one Iverson bracket.

There are an infinite variety of composite numbers.

The composite number trouble asks if there exist confident integers

and also
such that
.

A composite number

can constantly be written as a product in at least two ways (since
is always possible). Call these two products

(4)

then that is obviously the case that

(
divides
). Set

(5)

where

is the part of
, and also
is the component of
which divides
. Then there room
and also
such that

 (6) (7)

Solving

for
gives

(8)

It then follows that

 (9)(10)(11)

It because of this follows that

is never prime! In fact, the more general result that

(12)

is never prime for

an integer
=0" /> likewise holds (Honsberger 1991).

SEE ALSO: Amenable Number, Composite Number Problem, Grimm"s Conjecture, highly Composite Number, prime Factorization prime Gaps, element Number, Weakly Prime
REFERENCES:

Honsberger, R. An ext Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp.19-20, 1991.

Sloane, N.J.A. Succession A002808/M3272in "The On-Line Encyclopedia of integer Sequences."

Referenced top top sdrta.net|Alpha: Composite Number
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Weisstein, Eric W. "Composite Number."From sdrta.net--A sdrta.net net Resource. Https://sdrta.net/CompositeNumber.html

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