2025/11/19 by Louboutin, Stéphane
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Advanced Mathematical Identities
paper · doi:10.48550/arxiv.2511.15243
We fill the gaps in A. Gica's determination of all the odd positive integers d for which the number of distinct prime divisors of fd(x)=d+x2 is less than or equal to 2 for all the positive and odd integers x≤√(d). We also determine all the even positive integers d for which the number of distinct prime divisors of fd(x) is less than or equal to 2 for all the positive and even integers x≤√(d). These problems are related to the famous Frobenius-Rabinowitsch's characterization of the imaginary quadratic number fields \mathbb Q(√(-d)) of odd discriminants with class number one in terms of the primality of fd(x)/4 for all the positive and odd integers x≤√(d). However, the solution to our problem is much more difficult to come up with. We also begin to address the same problems for the case of fd(x)=d-x2, in relation with the class groups of the real quadratic number fields \mathbb Q(√(d)).