  In this paper we introduce and study the coprime quantum chain, i.e. a
strongly correlated quantum system defined in terms of the integer eigenvalues
$n_i$ of the occupation number operators at each site of a chain of length $M$.
The $n_i$'s take value in the interval $[2,q]$ and may be regarded as $S_z$
eigenvalues in the spin representation $j = (q-2)/2$. The distinctive
interaction of the model is based on the coprimality matrix $\bf \Phi$: for the
ferromagnetic case, this matrix assigns lower energy to configurations where
occupation numbers $n_i$ and $n_{i+1}$ of neighbouring sites share a common
divisor, while for the anti-ferromagnetic case it assigns lower energy to
configurations where $n_i$ and $n_{i+1}$ are coprime. The coprime chain, both
in the ferro and anti-ferromagnetic cases, may present an exponential number of
ground states whose values can be exactly computed by means of graph
theoretical tools. In the ferromagnetic case there are generally also
frustration phenomena. A fine tuning of local operators may lift the
exponential ground state degeneracy and, according to which operators are
switched on, the system may be driven into different classes of universality,
among which the Ising or Potts universality class. The paper also contains an
appendix by Don Zagier on the exact eigenvalues and eigenvectors of the
coprimality matrix in the limit $q \rightarrow \infty$.
