多项式的素数和素值

L’Enseignement Mathématique Pub Date : 2020-02-06 DOI:10.4171/lem/66-1/2-9

Arnaud Bodin, P. Dèbes, S. Najib

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引用次数: 6

摘要

Schinzel假设是数论中一个将多项式值和素数联系起来的著名猜想。以同样的方式，我们研究了几个多项式的值$P_1(n)，\ldots, P_s(n)$的公因数。我们推导出Schinzel假设的这个单素数版本:在一些自然的假设下，单素数多项式在无穷多个整数上假设单素数值。结果包括原始Schinzel假设的“取整数模”版本，以及哥德巴赫猜想，再次取整数模，作为特殊情况。

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Prime and coprime values of polynomials

The Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomial values and prime numbers. In the same vein we investigate the common divisors of values $P_1(n),\ldots, P_s(n)$ of several polynomials. We deduce this coprime version of the Schinzel Hypothesis: under some natural assumption, coprime polynomials assume coprime values at infinitely many integers. Consequences include a version "modulo an integer" of the original Schinzel Hypothesis, with the Goldbach conjecture, again modulo an integer, as a special case.

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L’Enseignement Mathématique

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