A possible connection between Mersenne primes and certain geometrical structures is implied. Here the authors consider the structures \((\mathbb{Z}_ q,{\mathcal B}_ p^ \#, \in)\) resulting from a planar nearring \((\mathbb{Z}_ q, +, *)\), where \(q= M_ p\) is a Mersenne prime, \(\mathbb{Z}_ q\) denotes the integers modulo \(q\), \(*\) is a multiplication of the Ferrero planar nearring factory manufactured by the subgroup of \(\mathbb{Z}_ q\) of order \(p\), \({\mathcal B}_ p^ \#\) is the set of circles \(C(r:b)= \mathbb{Z}_ p^* r+b\), \(r,b\in \mathbb{Z}_ q\), \(r\neq 0\), and \(\in\) is the usual element relation. It is shown that \((\mathbb{Z}_ q,{\mathcal B}_ p^ \#, \in)\) is circular, i.e. three distinct points of \(\mathbb{Z}_ q\) belong to at most one circle of \({\mathcal B}_ p^ \#\), and satisfies various properties which are not valid in the Euclidean plane and which have not been observed elsewhere. In particular, if \(k\) is the number of points on a circle \(C(r:0)\), then there are a maximum number, namely \(k-1\), of classes of circles \(E^ r_ b= \{C(r: b')\mid b'\in C(r:0)\}\) whose circles are tangent to \(C(r:0)\).