From its conception in the work of Kronecker and Hilbert until Takagi's great paper of 1920, class field theory was developed in order to prove reciprocity laws. When Artin succeeded in proving his reciprocity law in 1927, however, its role changed dramatically, and soon afterwards it became a cornerstone of class field theory. There are several natural formulations of Artin's reciprocity law involving ideals, ideles or cohomology groups. But can its content be made clear to ``any arithmetician from the street''? In this paper, the authors argue that this is the case. After discussing quadratic reciprocity as a special case of Artin's, the authors consider Mersenne primes \(M_p= 2^p-1\) for primes \(p\equiv 1\bmod 3\). These can be represented in the form \(M_p= x^2+ 7y^2\), and it is easily seen that we always have \(4\mid x\). Numerically, experiments suggest that in fact \(8\mid x\), and, amazingly, this can be proved using Artin's reciprocity law. The idea is that the congruence \(x\equiv 0\bmod 8\) is connected to the splitting of \(M_p\) in a certain subfield \(L\) of the ray class field modulo 8 of \(\mathbb{Q}(\sqrt{-7})\). Since this field \(L\) is also abelian over its subfield \(\mathbb{Q}(\sqrt{2})\), this splitting condition in \(\mathbb{Z} (\sqrt{-7})\) can be translated via Galois theory into a condition involving the factors of \(M_p\) in \(\mathbb{Z}(\sqrt{2})\). But the condition there can be verified directly, since \(M_p= (\sqrt{2}^p+1) (\sqrt{2}^p-1)\).