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 3 of 6: Test the cross factor

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In plain words

A small equation isolates the only exceptional parameter.

k_1k_2n+k_1+k_2=2n-3

Detailed analysis

If $(n-1)(2n-1)=(k_1n+1)(k_2n+1)$ with $k_1,k_2\ge1$ , then $k_1k_2n+k_1+k_2=2n-3$ . If either index is at least $2$ , the left side exceeds the right; if both are $1$ , equality occurs only for $n=5$ .