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Class numbers, Ono invariants and some interesting primes

Alexandru Gica

Abstract

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod 8$. The idea of the proof in these cases is to find the class number for the quadratic imaginary field $\mathbb{Q}(i\sqrt p)$. Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is $p\equiv 7 \pmod 8$. We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete.

Class numbers, Ono invariants and some interesting primes

Abstract

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases . The idea of the proof in these cases is to find the class number for the quadratic imaginary field . Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is . We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete.
Paper Structure (5 sections, 3 theorems, 52 equations)

This paper contains 5 sections, 3 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. The case $p\equiv 5 \pmod 8$
  3. The case $p\equiv 1 \pmod 8$
  4. The case $p\equiv 3 \pmod 8$
  5. The case $p\equiv 7 \pmod 8$

Key Result

Theorem 1.1

Let $p$ be an odd prime such that $p+x^2$ has at most two different prime factors for any odd integer $x$ such that $x^2<p$. If $p\equiv 5 \pmod 8$, then $p=5,13,37$. If $p\equiv 1 \pmod 8$, then $p=17,73,97,193$. If $p\equiv 3 \pmod 8$, then $p=3,11,19, 43, 67, 163$. If $p\equiv 7 \pmod 8$ then wit

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1