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 4 of 6: Rewrite the good-number condition as a norm

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In plain words

Working in the ring generated by a root of $T^2+T+k$ turns products of good numbers into norms of products of algebraic integers, which multiply nicely.

n^2+n+k=\operatorname{Norm}(n-\alpha)

Detailed analysis

Fix $\alpha\in\mathbb{C}$ with $\alpha^2+\alpha+k=0$ , and for $n\in\mathbb{Z}[\alpha]$ define $\operatorname{Norm}(n)=n\bar n$ where $\bar\alpha=-1-\alpha$ is the conjugate root. Then for any integer $n$ , $n^2+n+k=\operatorname{Norm}(n-\alpha)$ , so $n$ is good exactly when $n=\operatorname{Norm}(n-\alpha)$ for an algebraic integer $n-\alpha$ of this special form.