منابع مشابه
Artin prime producing polynomials
Abstract. We define an Artin prime for an integer g to be a prime such that g is a primitive root modulo that prime. Let g ∈ Z \ {−1} and not be a perfect square. A conjecture of Artin states that the set of Artin primes for g has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of ...
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where each <p,(z) is a polynomial of degree greater than unity, prime or composite, we shall say that (2) is a decomposition of F(z). The first result of the present paper is that any two decompositions of a given polynomial into prime polynomials contain the same number of polynomials; the degrees of the polynomials in one decomposition are the same as those in the other, except, perhaps, for ...
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Some nonconstant polynomials with a finite string of prime values are known; in this paper, some polynomials of this kind are described, starting from Euler’s example (1772) P(x) = x2+x+41: other quadratic polynomials with prime values were studied, and their properties were related to properties of quadratic fields; in this paper, some quadratic polynomials with prime values are described and ...
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The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 (see Lang [3, p. 323]), independently reformulated by Schinzel, to the effect that ...
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This article provides necessary and sufficient conditions for a real quadratic field to have class number one or two in terms of a new set of primeproducing quadratic polynomials
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2014
ISSN: 0025-5718,1088-6842
DOI: 10.1090/s0025-5718-2014-02902-5