[Paper Review] Primitive root producing quadratics

Pieter Moree|ArXiv.org|Jun 2, 2004

Analytic Number Theory Research14 references3 citations

TL;DR

This paper investigates quadratic polynomials that produce primes for which a fixed integer g is a primitive root, linking this to class number one problems and prime-producing polynomials. It proves the existence of such quadratics with arbitrarily long sequences of consecutive primes for which g is a primitive root, using class field theory and the prime k-tuplets conjecture, and establishes that 326 is a primitive root modulo the first 206 primes of the form 326n² + 3, with Y. Gallot later extending this to 31,082 primes.

ABSTRACT

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. An algorithm is described to find more impressive examples in the same spirit. Y. Gallot used it to establish the current record in which 206 is being replaced by 31082.

Motivation & Objective

• To investigate the phenomenon where a fixed integer g is a primitive root modulo many consecutive primes generated by a quadratic polynomial f(n).
• To connect this behavior to the class number one problem for imaginary quadratic fields and to prime-producing polynomials.
• To demonstrate that for any given m, there exist quadratic polynomials f and integers g such that g is a primitive root modulo the first m primes represented by f(n).
• To formalize the concept of 'primitive root producing' polynomials and analyze their theoretical limits.

Proposed method

• Uses the Bateman-Horn and Hardy-Littlewood Conjectures to model the density of primes represented by quadratic polynomials.
• Applies class field theory and quadratic reciprocity to control the splitting behavior of primes in number fields Q(√g_i).
• Employs the prime k-tuplets conjecture to ensure the simultaneous primality of multiple values of f(n) in arithmetic progression.
• Constructs explicit quadratic polynomials f(n) = An² + C such that g is a primitive root modulo f(n) for all n in a given range where f(n) is prime.
• Uses the Chinese Remainder Theorem and equidistribution assumptions to estimate the size of the smallest solution satisfying primitive root conditions modulo a set of primes.
• Leverages results on L-functions (L(1,χ_D) and L(2,χ_D)) to relate the analytic behavior of Dirichlet L-functions to the existence of such polynomials.

Experimental results

Research questions

• RQ1 Can a fixed integer g be a primitive root modulo a long sequence of consecutive primes generated by a quadratic polynomial f(n)?
• RQ2 What number-theoretic conditions on g and f ensure that g is a primitive root modulo all primes f(n) that are prime and do not divide g?
• RQ3 Is there a theoretical upper bound on the length of such sequences, or can they be arbitrarily long?
• RQ4 How are primitive root producing polynomials related to the class number one problem for imaginary quadratic fields?
• RQ5 To what extent can the prime k-tuplets conjecture be used to construct such polynomials with guaranteed long sequences of primitive root primes?

Key findings

• The paper proves that for any integer m ≥ 1, there exist quadratic polynomials f and integers g such that g is a primitive root modulo the first m primes of the form f(n) that are prime and do not divide g.
• The example f(n) = 326n² + 3 yields 206 consecutive primes for which 326 is a primitive root, a result tied to the class number one of Q(√−163).
• Y. Gallot extended this record to 31,082 consecutive primes using similar principles and advanced computation.
• The existence of such polynomials is conditional on the prime k-tuplets conjecture and the generalized Riemann hypothesis, but the construction is theoretically sound under these assumptions.
• The paper shows that the density of such polynomials is governed by the size of L(2,χ_D), analogous to how L(1,χ_D) governs prime-producing behavior.
• It is proven that no quadratic polynomial can produce infinitely many primes for which g is a primitive root, i.e., c_g(f) < ∞, but arbitrarily long finite sequences are possible.