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 6 of 6: Handle

n=5

and

n=8

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In plain words

Explicit witnesses cover the exceptional cases.

3136=16\cdot196=56\cdot56,\quad25921=49\cdot529=161\cdot161

Detailed analysis

For $n=5$ , $16,196,56$ are indecomposable elements of $V_5$ and $16\cdot196=56\cdot56$ . For $n=8$ , $49,529,161$ are indecomposable elements of $V_8$ and $49\cdot529=161\cdot161$ . Each identity gives two distinct factorizations.