1969/04/01 by H. Stärk · 2 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #Discriminant #Class number #Mathematics #Prime (order theory) #Combinatorics #Quadratic equation #Field (mathematics) #Kronecker delta #Algebraic number field #Class (philosophy) #Quadratic form (statistics) #Quadratic field #Symbol (formal) #Prime number #Discrete mathematics #Arithmetic #Pure mathematics #Physics #Quadratic function #Computer science #Geometry #Quantum mechanics #Artificial intelligence
paper · pdf · doi:10.1090/s0002-9939-1969-0237461-x
openalex publication_date 1969/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
Let d be the discriminant of a quadratic field, h(d) the class-number of the field and Xd the character (mod I dj) given by Xd(n) = (d/n) (Kronecker symbol). If we restrict d to values <-8 with h(d) = 1 then I dj is a prime3 (mod 8) and it has recently been proved [I], [4] that d ? 163. The purpose of this note is to show that Gelfond and Linnik missed an opportunity to prove this result in 1949. Let k be the discriminant of a real quadratic field and Ec the fundamental unit of Q(x/k). In the sequel, we assume d < -8 and h(d) = 1. The following expansion was known for prime k by Heilbronn and Linfoot [31 and has recently been established [5] for all k such that (k, d) = 1,