[Paper Review] Squares in Polynomial Product Sequences

Paul Spiegelhalter, Joseph Vandehey|arXiv (Cornell University)|Jul 8, 2011

Analytic Number Theory Research11 references3 citations

TL;DR

This paper investigates how frequently the product sequence $ N_x = \prod_{n \leq x} F(n) $, where $ F(n) $ is a polynomial of degree at least 2 with integer coefficients, is a perfect power or squarefull number. Using analytic number theory, including the Turán sieve and bounds on Chebotarev density, the authors show that such products are perfect $ p $-th powers for only $ O(X^c) $ values of $ x \leq X $ with $ c < 1 $, and provide explicit bounds for monic irreducible quadratics when $ N_x $ is squarefull.

ABSTRACT

Let F(n) be a polynomial of degree at least 2 with integer coefficients. We consider the products N_x=\prod_{1 \le n \le x} F(n) and show that N_x should only rarely be a perfect power. In particular, the number of x \le X for which N_x is a perfect power is O(X^c) for some explicit c&lt;1. For certain F(n) we also prove that for only finitely many x will N_x be squarefull and, in the case of monic irreducible quadratic F(n), provide an explicit bound on the largest x for which N_x is squarefull.

Motivation & Objective

• To determine how often the product $ N_x = \prod_{n \leq x} F(n) $ is a perfect power or squarefull for polynomials $ F(n) $ of degree ≥2.
• To extend prior results on specific polynomials (e.g., $ n^2+1 $) to broader classes of polynomials.
• To establish explicit upper bounds on the number of $ x \leq X $ for which $ N_x $ is a perfect power or squarefull.
• To analyze the structure of $ F(n) $, particularly when factored into irreducible components, to determine finiteness of squarefull occurrences.
• To provide effective, computable bounds for the largest $ x $ such that $ N_x $ is squarefull in the monic irreducible quadratic case.

Proposed method

• Applies the Turán sieve to estimate the number of $ x \leq X $ for which $ F_k(x) = \prod_{i=1}^k F(x+i) $ is a perfect $ p $-th power.
• Uses bounds on the Chebotarev density theorem and character sum estimates to control the distribution of primes modulo which $ F_k(n) $ is a $ p $-th power.
• Employs the Brun-Titchmarsh theorem and Pólya-Vinogradov type estimates to bound sums over arithmetic progressions of primes.
• Derives effective bounds on $ \sum_{p \leq z, p \equiv a \pmod{q}} \frac{\log p}{p} $ using explicit constants from the prime number theorem in arithmetic progressions.
• Applies a double counting argument via Lemma 5.21 to show that if many $ x $ yield perfect $ p $-th power $ N_x $, then some $ F_k(n) $ must be a $ p $-th power for many $ n $, contradicting sieve estimates.
• Uses the structure of discriminants and splitting types of irreducible factors to constrain the number of $ x $ for which $ N_x $ is squarefull.

Experimental results

Research questions

• RQ1 For a polynomial $ F(n) $ of degree ≥2 with integer coefficients, how often is the product $ N_x = \prod_{n \leq x} F(n) $ a perfect $ p $-th power?
• RQ2 Under what conditions on $ F(n) $, particularly when factored into irreducible components, is $ N_x $ squarefull for only finitely many $ x $?
• RQ3 Can explicit upper bounds be given for the largest $ x $ such that $ N_x $ is squarefull, especially when $ F(n) $ is a monic irreducible quadratic?
• RQ4 How does the number of $ x \leq X $ for which $ N_x $ is a perfect power grow as a function of $ X $, and can this growth be bounded by $ O(X^c) $ with $ c < 1 $?
• RQ5 What is the role of the leading coefficient and discriminant of irreducible factors in determining the frequency of perfect power or squarefull values of $ N_x $?

Key findings

• For any polynomial $ F(n) $ of degree ≥2 with integer coefficients, the number of $ x \leq X $ such that $ N_x $ is a perfect $ p $-th power is $ O(X^c) $ for some explicit $ c < 1 $, independent of $ F $.
• If $ F(n) $ is not of the form $ sG(n)^p $ for rational $ s $ and $ G(n) \in \mathbb{Z}[n] $, then $ N_x $ is a perfect $ p $-th power for only $ O(X^c) $ values of $ x \leq X $, with $ c < 1 $.
• For monic irreducible quadratic $ F(n) $, an explicit upper bound on the largest $ x $ for which $ N_x $ is squarefull is provided, specifically $ O(e^{C \cdot D}) $, where $ D $ is the discriminant and $ C $ is effectively computable.
• If $ F(n) $ factors into linear and quadratic terms with certain conditions on leading coefficients and discriminants, then $ N_x $ is squarefull for only finitely many $ x $, covering large families such as products of two or three distinct irreducible quadratics.
• The method yields effective bounds for the case $ F(n) = n^2 + D $, showing that $ N_x $ is squarefull only for $ x \leq e^{C \cdot D} $, with $ C $ computable.
• The analysis shows that the number of $ x \leq X $ for which $ N_x $ is a perfect $ p $-th power is bounded by $ O(X^{1/25}) $ in the worst case, derived from sieve and counting arguments.