Problem 3

Let

k

be a positive integer and let

S

be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of

S

around a circle such that the product of any two neighbours is of the form

x^2+x+k

for some positive integer

x

Step 1 of 6: Call a number good and allow

x\ge0

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In plain words

The substitution $x\mapsto-1-x$ leaves $x^2+x+k$ unchanged, so restricting to positive $x$ or nonnegative $x$ describes the same set of values.

n\ \text{good}\iff n=x^2+x+k,\ x\in\mathbb{Z}_{\ge0}

Detailed analysis

Since $(-1-x)^2+(-1-x)+k=x^2+x+k$ , calling $n$ good when $n=x^2+x+k$ for some nonnegative integer $x$ describes exactly the same numbers as requiring a positive integer $x$ ; we use the nonnegative convention for convenience.