Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A problem of Polya

Umberto Zannier

TL;DR

The paper analyzes a non-constant-coefficient linear recurrence tied to a carefully chosen rational sequence, uncovering unexpected $p$-adic congruences and denominators that are powers of 2. It reveals that the generating function is algebraic and intimately connected to an elliptic curve, enabling a p-adic and differential-analytic treatment via the Cartier operator and logarithmic derivatives. By leveraging Grothendieck’s conjecture, Honda–Katz theory, and the Chudnovski–André theorem, the authors establish congruence properties (Theorem T.congr) and a converse result (Theorem T.converse), and they relate these to integrality and p-curvature phenomena. The work ties Recurrence-Arithmetic to algebraic geometry, using explicit elliptic-curve models to illustrate and prove the key congruences, while discussing Polya-type questions and density issues for primes, with several open questions and CM-specific observations.

Abstract

We prove an improved form of an expectation of Polya and discuss several related questions

A problem of Polya

TL;DR

The paper analyzes a non-constant-coefficient linear recurrence tied to a carefully chosen rational sequence, uncovering unexpected -adic congruences and denominators that are powers of 2. It reveals that the generating function is algebraic and intimately connected to an elliptic curve, enabling a p-adic and differential-analytic treatment via the Cartier operator and logarithmic derivatives. By leveraging Grothendieck’s conjecture, Honda–Katz theory, and the Chudnovski–André theorem, the authors establish congruence properties (Theorem T.congr) and a converse result (Theorem T.converse), and they relate these to integrality and p-curvature phenomena. The work ties Recurrence-Arithmetic to algebraic geometry, using explicit elliptic-curve models to illustrate and prove the key congruences, while discussing Polya-type questions and density issues for primes, with several open questions and CM-specific observations.

Abstract

We prove an improved form of an expectation of Polya and discuss several related questions

Paper Structure

This paper contains 20 sections, 20 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Comments on the results.
  3. Some congruences modulo $p$ for coefficients of power series
  4. A deduction of Fermat's Little Theorem.
  5. A converse proposition.
  6. An elliptic example.
  7. An amusing application to a proof of Theorem \ref{['T.congr']}
  8. An elliptic curve.
  9. An 'explicit' formula.
  10. Changing the initial data and a proof of Theorem \ref{['T.converse']} (i)
  11. Solving by 'quadratures'.
  12. A question of Pòlya and its evolutions
  13. A result of Honda and Katz
  14. A theorem of Chudnovski and André and a proof of Theorem \ref{['T.converse']} (ii)
  15. The Cartier operator
  16. ...and 5 more sections

Key Result

Theorem 1.1

The $c_n$ do not satisfy any linear recurrence with constant coefficients. They are rational numbers whose denominator is a power of $2$. Moreover, for any odd prime $p$, they satisfy the congruences

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 42 more