MathLabs

Problem 3

Let kk be a positive integer and let SS 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 SS around a circle such that the product of any two neighbours is of the form x2+x+kx^2+x+k for some positive integer xx.
Step 5 of 6: Show qrqr is also good when pq,prpq,pr both are
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(x−α)(y−α)=p(z−α),z=xy−kp,qr=z2+z+k(x-\alpha)(y-\alpha)=p(z-\alpha),\quad z=\frac{xy-k}{p},\quad qr=z^2+z+k
Detailed analysis

Choose the same root α\alpha. Then (x−α)(y−α)=xy−(x+y)α+α2=xy−k−(x+y+1)α=p(z−α)(x-\alpha)(y-\alpha)=xy-(x+y)\alpha+\alpha^2=xy-k-(x+y+1)\alpha=p(z-\alpha), where z=(xy−k)/p∈Zz=(xy-k)/p\in\mathbb Z by xy≡k(modp)xy\equiv k\pmod p. Taking norms gives p2qr=p2(z2+z+k)p^2qr=p^2(z^2+z+k), hence qr=z2+z+kqr=z^2+z+k is good (if z<0z<0, replace it by −1−z-1-z).