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 2 of 6: Prove the first factor indecomposable

In plain words

The smallest possible product gives an immediate indecomposability test.

(n-1)^2<(n+1)^2

Detailed analysis

Every product of two members of $V_n$ is at least $(n+1)^2$ . Since $(n-1)^2$ is smaller, it cannot be decomposable.