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Are prime numbers and quadratic residues random?

Michael Blank

TL;DR

The paper tackles whether prime numbers and quadratic residues exhibit true randomness by introducing trajectory-local entropy notions $h_{loc}$ and $h_{info}$ that quantify complexity of a single deterministic sequence independent of invariant measures. It develops a framework connecting static Shannon entropy to dynamic, trajectory-focused measures and applies it to number-theoretic objects. The results show that the binary prime-indicator sequence has low trajectory entropy (bounded by $\log\frac{1+\sqrt{5}}{2}$ and, under Hardy–Littlewood, attains this bound), while the quadratic-residue sequence exhibits maximal trajectory randomness with $h_{loc}=h_{info}=1$. The analysis also links to inhomogeneous Bernoulli models and Cramér’s prime model, where the inhomogeneous case yields vanishing trajectory entropy, highlighting a nuanced spectrum of randomness in deterministic number-theoretic structures. Overall, the work offers a quantitative, trajectory-centered lens for distinguishing randomness in primes versus quadratic residues and provides tools for analyzing single sequences beyond classical system-wide entropy.

Abstract

Appeals to randomness in various number-theoretic constructions appear regularly in modern scientific publications. Such famous names as V.I. Arnold, M. Katz, Ya.G. Sinai, and T. Tao are just a few examples. Unfortunately, all of these approaches rely on various, although often very non-trivial and elegant, heuristics. A new analytical approach is proposed to address the issue of randomness/complexity of an individual deterministic sequence. This approach demonstrates the expected high complexity of quadratic residues and the unexpectedly low complexity in the case of prime numbers. Technically, our approach is based on a new construction of the dynamical entropy of a single trajectory, which measures its complexity, in contrast to classical Kolmogorov-Sinai and topological entropies, which measure the complexity of the entire dynamical system.

Are prime numbers and quadratic residues random?

TL;DR

The paper tackles whether prime numbers and quadratic residues exhibit true randomness by introducing trajectory-local entropy notions and that quantify complexity of a single deterministic sequence independent of invariant measures. It develops a framework connecting static Shannon entropy to dynamic, trajectory-focused measures and applies it to number-theoretic objects. The results show that the binary prime-indicator sequence has low trajectory entropy (bounded by and, under Hardy–Littlewood, attains this bound), while the quadratic-residue sequence exhibits maximal trajectory randomness with . The analysis also links to inhomogeneous Bernoulli models and Cramér’s prime model, where the inhomogeneous case yields vanishing trajectory entropy, highlighting a nuanced spectrum of randomness in deterministic number-theoretic structures. Overall, the work offers a quantitative, trajectory-centered lens for distinguishing randomness in primes versus quadratic residues and provides tools for analyzing single sequences beyond classical system-wide entropy.

Abstract

Appeals to randomness in various number-theoretic constructions appear regularly in modern scientific publications. Such famous names as V.I. Arnold, M. Katz, Ya.G. Sinai, and T. Tao are just a few examples. Unfortunately, all of these approaches rely on various, although often very non-trivial and elegant, heuristics. A new analytical approach is proposed to address the issue of randomness/complexity of an individual deterministic sequence. This approach demonstrates the expected high complexity of quadratic residues and the unexpectedly low complexity in the case of prime numbers. Technically, our approach is based on a new construction of the dynamical entropy of a single trajectory, which measures its complexity, in contrast to classical Kolmogorov-Sinai and topological entropies, which measure the complexity of the entire dynamical system.
Paper Structure (7 sections, 19 theorems, 45 equations)

This paper contains 7 sections, 19 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Information (Shannon) entropy
  3. Dynamical entropy in ergodic theory
  4. Complexity
  5. Spatial distribution of prime numbers
  6. Inhomogeneous random Bernoulli process and Cramer's model
  7. Spatial distribution of quadratic residues

Key Result

Lemma 2.1

Let $r:=\#\Omega\le\infty$. Then and this functional reaches its maximum value on the uniform distribution.

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.1
  • ...and 27 more