Problem 3

Let

n>2

and let

V_n=\{1+kn:k=1,2,\ldots\}

. An element of

V_n

is indecomposable if it is not a product of two elements of

V_n

. Prove that some element of

V_n

has two distinct factorizations into indecomposable elements, where order is ignored.

Step 5 of 6: Finish the generic case

In plain words

Pairing four numbers in two ways gives distinct factor multisets.

r=(n-1)^2(2n-1)^2

Detailed analysis

For $n\ne5,8$ , all three displayed factors are indecomposable, so their product has the two distinct indecomposable factorizations above.