# Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm

Source: [Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm](<https://www.sota2.com/research/paper/68fa5d781e209131ce9da5ff>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T18:16:09.851Z

Authors: K. Lakshmanan

Published: 2024-05-17T03:57:40.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2405.10519>)

[PDF](<https://arxiv.org/pdf/2405.10519.pdf>)