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Composite Number -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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A composite number n is a positive integer n>1 which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called "composites" for short) are 4, 6, 8, 9, 10, 12, 14, 15, 16, ... (OEIS A002808), whose prime decompositions are summarized in the following table. Note that the number 1 is a special case which is considered to be neither composite nor prime.

The nth composite number c_n can be generated using the Wolfram Language code 

  Composite[n_Integer] :=
    FixedPoint[n + PrimePi[#] + 1&, n]

The Dirichlet generating function of the characteristic function of the composite numbers c_n is given by

where zeta(s) is the Riemann zeta function, P(s) is the prime zeta function, and [S] is an Iverson bracket.

There are an infinite number of composite numbers.

The composite number problem asks if there exist positive integers m and n such that N=mn.

A composite number C can always be written as a product in at least two ways (since 1·C is always possible). Call these two products

C=ab=cd,

(4)

then it is obviously the case that c|ab (c divides ab). Set

c=mn,

(5)

where m is the part of c which divides a, and n is the part of c which divides b. Then there are p and q such that

Solving ab=cd for d gives

d=(ab)/c=((mp)(nq))/(mn)=pq.

(8)

It then follows that

It therefore follows that a^2+b^2+c^2+d^2 is never prime! In fact, the more general result that

S=a^k+b^k+c^k+d^k

(12)

is never prime for k an integer >=0 also holds (Honsberger 1991).

See also

Amenable Number, Composite Number Problem, Grimm's Conjecture, Highly Composite Number, Prime Factorization Prime Gaps, Prime Number, Weakly Prime

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References

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 19-20, 1991.Sloane, N. J. A. Sequence A002808/M3272 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Composite Number

Cite this as:

Weisstein, Eric W. "Composite Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompositeNumber.html

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