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Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm

K. Lakshmanan

TL;DR

This work tackles the problem of whether a bivariate polynomial $f(x,y)$ with non-negative coefficients and no constant term can take a prime value. It introduces a simple, constructive recursive obstruction based on gcd-divisibility patterns from substitutions, which is finite to verify up to $f(2,2)$ and yields a sufficient condition for non-primality. The main contribution is an algorithmic criterion that, if satisfied for all small input tuples, guarantees that $f(x,y)$ represents no primes; examples illustrate both prime-producing and prime-free cases, and the approach provides a perspective that can extend to multivariate and finite-field settings. The results complement classical conjectures like Bouniakowsky by offering a practical method to certify the absence of prime outputs in a broad class of bivariate polynomials and by pointing to several natural directions for future work and generalizations.

Abstract

We investigate the computational problem of determining whether a bivariate polynomial with non-negative coefficients and no constant term can attain a prime value. While classical conjectures such as Bouniakowsky's provide necessary conditions for univariate prime-representing polynomials, we introduce a new recursive algorithm that efficiently certifies when a bivariate polynomial form can produce no prime values at all. Our method is elementary and constructive, based on analyzing gcd-divisibility patterns arising from recursive substitutions into the polynomial. The obstruction criterion obtained leads to an efficient and elementary algorithm that certifies when a polynomial form cannot produce any prime values. The result is stronger than what is implied by the negation of Bouniakowsky's condition and applies to a wide class of polynomials, including transformations of univariate forms. We provide illustrative examples, analyze the complexity of the method, and discuss its connections to existing conjectures and possible generalizations.

Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm

TL;DR

This work tackles the problem of whether a bivariate polynomial with non-negative coefficients and no constant term can take a prime value. It introduces a simple, constructive recursive obstruction based on gcd-divisibility patterns from substitutions, which is finite to verify up to and yields a sufficient condition for non-primality. The main contribution is an algorithmic criterion that, if satisfied for all small input tuples, guarantees that represents no primes; examples illustrate both prime-producing and prime-free cases, and the approach provides a perspective that can extend to multivariate and finite-field settings. The results complement classical conjectures like Bouniakowsky by offering a practical method to certify the absence of prime outputs in a broad class of bivariate polynomials and by pointing to several natural directions for future work and generalizations.

Abstract

We investigate the computational problem of determining whether a bivariate polynomial with non-negative coefficients and no constant term can attain a prime value. While classical conjectures such as Bouniakowsky's provide necessary conditions for univariate prime-representing polynomials, we introduce a new recursive algorithm that efficiently certifies when a bivariate polynomial form can produce no prime values at all. Our method is elementary and constructive, based on analyzing gcd-divisibility patterns arising from recursive substitutions into the polynomial. The obstruction criterion obtained leads to an efficient and elementary algorithm that certifies when a polynomial form cannot produce any prime values. The result is stronger than what is implied by the negation of Bouniakowsky's condition and applies to a wide class of polynomials, including transformations of univariate forms. We provide illustrative examples, analyze the complexity of the method, and discuss its connections to existing conjectures and possible generalizations.
Paper Structure (14 sections, 9 theorems, 16 equations)

This paper contains 14 sections, 9 theorems, 16 equations.

Table of Contents

  1. Introduction and Motivation
  2. Background and Related Work
  3. Definitions and Setup
  4. Main Result
  5. Proof of Theorem \ref{['mth']}
  6. Algorithm and Constructive Condition
  7. Examples and Case Studies
  8. Example 1: A Prime-Producing Polynomial
  9. Example 2: A Non-Prime-Producing Polynomial
  10. Example 3: Another Non-Prime Producing Polynomial
  11. Comparison with Bouniakowsky Conjecture
  12. Deeper Remarks and Extensions
  13. Open Problems and Future Work
  14. Conclusion

Key Result

Lemma 1

Let $f(x, y) = \sum_{i+j>0} k_{ij} x^i y^j$ be a bivariate polynomial with $k_{ij} \in \mathbb{Z}_{\geq 0}$ and $k_{00} = 0$. Then $f(x, y) > 0$ for all $x, y \in \mathbb{Z}^+$.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof : Sketch of Idea
  • Theorem 2
  • Remark 1: Minimal Recursive Obstruction Hypothesis
  • Theorem 3
  • Lemma 3
  • ...and 7 more