Euler's famous prime generating polynomial and the class number of imaginary quadratic fields
Are there infinitely many, or only finitely many primes q such that Euler's generalized polynomial \(f_ q(X)=X^ 2+X+q\) has a prime value for \(X=0,1,...,q-2?\) What is the largest possible q? In this connection, the author proves that the following three conditions are equivalent: \((1)\quad q=2,3,5,11,17,41;\) \((2)\quad f_ q(X)\) is a prime for \(X=0,1,2,...,q-2\); \((3)\quad the\) imaginary quadratic field \({\mathbb{Q}}(\sqrt{1-4q})\) has class number one. This lecture also contains some facts from the elementary theory of quadratic fields (units, class number, theorem of Rabinovitch etc.) with a historical excursion.
scientific article; zbMATH DE number 4083760