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 4 of 6: Test the second square

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In plain words

The size estimate isolates $n=8$ as the one exceptional parameter for this square.

k_1k_2n+k_1+k_2=4n-4

Detailed analysis

A decomposition of $(2n-1)^2$ gives $k_1k_2n+k_1+k_2=4n-4$ . For $n>2$ , $(1,1)$ would require $n=2$ , $(1,2)$ would require $n=7/2$ , and $(1,3)$ gives exactly $n=8$ . Every other positive pair makes the left side exceed the right.