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 1 of 6: Choose four numbers in the

-1

class

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In plain words

Multiplying two numbers congruent to $-1$ returns to the semigroup $V_n$ .

n-1\equiv2n-1\equiv-1\pmod n

Detailed analysis

The numbers $(n-1)^2$ , $(2n-1)^2$ , and $(n-1)(2n-1)$ are all in $V_n$ , since their residues are $1\pmod n$ . The product of the first two equals the square of the third: $(n-1)^2(2n-1)^2=((n-1)(2n-1))^2$ .