We have m = ab = cd, where ab and cd are distinct factorizations, and a, b, c, d 1.
Clearly ab is a multiple of c.
By the Fundamental Theorem of Arithmetic, each prime factor of c occurs in a or b.
Let c = rs, where r is the product of the prime factors of c that occur in a, and s is the product of the prime factors of c that occur in b.
(In the absence of any common prime factors, r or s, or both, may equal 1.)
Then there exist u and v (possibly equal to 1) such that:
a = ru
b = sv
We also have d = ab/c.
Hence d = (ru)(sv)/(rs) = uv.
|an + bn + cn + dn||= rnun + snvn + rnsn + unvn|
|= (rn + vn)(sn + un)|
Both factors are greater than 1, for all integers n 0.
Therefore an + bn + cn + dn is composite.