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Suppose that $q$ is prime. We construct a graph as follows: the vertices are prime divisors of the numbers $(q-1)/2$, $(q+1)/2$ and $(q^{2}-1)/24$.

Two vertices $r$ and $s$ are joined by an edge if $rs$ divides some of the numbers $(q-1)/2$, $(q+1)/2$ or $(q^{2}-1)/24$.

Is it true this graph is disconnected?

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$\frac{q^2 - 1}{24} = \frac{1}{6} \frac{q - 1}{2} \cdot \frac{q + 1}{2}$ should help... –  vonbrand Feb 23 '13 at 14:54
    
@user57106 are you looking to characterize which values of $q$ for which the graph is connected? Otherwise, one can consider the following examples: $q=13$, which leads to a disconnected graph with vertices $2,3,7$, and $q-23$, which leads to a connected graph with vertices $2,3,11$. –  Vincent Tjeng Feb 25 '13 at 5:51

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