Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918034442027008 |
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| author | Lakshmanan, K. |
| author_facet | Lakshmanan, K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm Lakshmanan, K. Number Theory Primary 11A41, Secondary 11C08 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. |
| title | Primes and Bivariate Polynomials without Constant Terms: A Recursive Algorithm |
| topic | Number Theory Primary 11A41, Secondary 11C08 |
| url | https://arxiv.org/abs/2405.10519 |