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Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials

Stéphane Louboutin

TL;DR

The article builds on and corrects gaps in GicaIndMath to advance the understanding of how many distinct prime divisors occur in values of quadratic polynomials tied to quadratic fields. It combines elementary arguments with algebraic number theory, using class-number theory for imaginary quadratic fields and refined Minkowski-type bounds to constrain possible d with M_odd(d) ≤ 2 or M_even(d) ≤ 2, including detailed treatments for d ≡ 1,3,5,7 (mod 8) and the special case d ≡ 2 (mod 4). The work yields explicit finite lists of d in many cases (both prime and composite forms) and establishes bounds on class numbers (often divisors of small powers of 2), while also beginning to address the corresponding real quadratic problems. The results have implications for Frobenius–Rabinowitsch-type characterizations and provide a framework for further refinement of the “d-not-prime-not-qpq” landscape, including conjectural finite sets in the real case and several not-yet-complete directions for square-free and non-square-free d.

Abstract

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 $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also determine all the even positive integers $d$ for which the number of distinct prime divisors of $f_d(x)$ is less than or equal to $2$ for all the positive and even integers $x\leq\sqrt{d}$. These problems are related to the famous Frobenius-Rabinowitsch's characterization of the imaginary quadratic number fields ${\mathbb Q}(\sqrt{-d})$ of odd discriminants with class number one in terms of the primality of $f_d(x)/4$ for all the positive and odd integers $x\leq\sqrt{d}$. 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 $f_d(x)=d-x^2$, in relation with the class groups of the real quadratic number fields ${\mathbb Q}(\sqrt{d})$.

Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials

TL;DR

The article builds on and corrects gaps in GicaIndMath to advance the understanding of how many distinct prime divisors occur in values of quadratic polynomials tied to quadratic fields. It combines elementary arguments with algebraic number theory, using class-number theory for imaginary quadratic fields and refined Minkowski-type bounds to constrain possible d with M_odd(d) ≤ 2 or M_even(d) ≤ 2, including detailed treatments for d ≡ 1,3,5,7 (mod 8) and the special case d ≡ 2 (mod 4). The work yields explicit finite lists of d in many cases (both prime and composite forms) and establishes bounds on class numbers (often divisors of small powers of 2), while also beginning to address the corresponding real quadratic problems. The results have implications for Frobenius–Rabinowitsch-type characterizations and provide a framework for further refinement of the “d-not-prime-not-qpq” landscape, including conjectural finite sets in the real case and several not-yet-complete directions for square-free and non-square-free d.

Abstract

We fill the gaps in A. Gica's determination of all the odd positive integers for which the number of distinct prime divisors of is less than or equal to for all the positive and odd integers . We also determine all the even positive integers for which the number of distinct prime divisors of is less than or equal to for all the positive and even integers . These problems are related to the famous Frobenius-Rabinowitsch's characterization of the imaginary quadratic number fields of odd discriminants with class number one in terms of the primality of for all the positive and odd integers . 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 , in relation with the class groups of the real quadratic number fields .

Paper Structure

This paper contains 14 sections, 31 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['dnotprimenotpq']}
  3. Notations
  4. Proof of Theorem \ref{['mainthimpairnot7mod8']}
  5. The case $d\equiv 3\pmod 8$ is prime
  6. The case $d\equiv 5\pmod 8$ is prime
  7. The case $d\equiv 1\pmod 8$ is prime
  8. Proof of Theorem \ref{['mainthimpair7mod8']}
  9. Proof of Theorem \ref{['mainthd=2p']}
  10. Proof of Theorem \ref{['mainthd=2mod4']}
  11. Proof of Theorem \ref{['mainthd=2mod4notsquarefree']}
  12. Proof of Theorem \ref{['mainthd=2mod4reel']}
  13. Proof of Theorem \ref{['mainthd=2mod4notsquarefreereel']}
  14. Concluding remarks

Key Result

Theorem 1

Let $d\geq 1$ be odd, neither prime nor the product of two distinct primes. Then $M_{odd}(d)\leq 2$ if and only if $d\in\{1, 9, 25, 27, 49, 63, 135, 175, 207, 343\}$ ($10$ values).

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Conjecture 7
  • Conjecture 8
  • Theorem 9
  • Conjecture 10
  • ...and 48 more