Densities of Sets of Primes Related to Decimal Expansion of Rational Numbers

Hadano, Toshihiro; Kitaoka, Yoshiyuki; Nozaki, Michihiro

doi:10.1007/0-387-30829-6_6

Cited by 5 publications

(12 citation statements)

References 2 publications

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“…For a prime number p which decomposes fully in F and a prime ideal p lying above p, we write in Indeed, we can show that Conjecture 3.1 yields the assertion on the average as follows. The proof is quite similar to the proof of Proposition 1 in [1]. The following gives a connection between Conjecture 4 in [1] and the viewpoint in this paper.…”

Section: Miscellaneous Remarkssupporting

confidence: 63%

A statistical relation of roots of a polynomial in different local fields

Kitaoka

2009

Math. Comp.

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Section: Miscellaneous Remarkssupporting

confidence: 63%

A statistical relation of roots of a polynomial in different local fields

Kitaoka

2009

Math. Comp.

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“…We checked densities Pr X (f, 2, {R i }) approximate well densities of special polynomials 11 ) approximates well densities given above and the densities on p.87 in [2] should be corrected by the above.…”

Section: Distribution Of Roots Modulo a Primementioning

confidence: 96%

“…where local roots r i satisfy properties (1), (2). We expect, under an assumption that a polynomial f has only a trivial linear relation (3) among roots.…”

Section: Communicated By Shigeki Akiyamamentioning

confidence: 99%

“…The two conditions irreducibility and indecomposability are equivalent to having no non-trivial linear relations among roots in the case of degree ≤ 5 as in the previous section. Data in [2] are less accurate in the case of degree 6, that is X in (11) was too small to guess the precise limit. We improve a method to guess the limit from approximate values.…”

Section: Expectation 2 For a Polynomial Of Degreementioning

confidence: 99%

“…(I) The case that there is no non-trivial linear relation among roots : For a polynomial f = x 6 + 5x 5 + 1 in [2], data were insufficient. Wrong values 7/512 = 0.0136 .…”

Section: Expectation 2 For a Polynomial Of Degreementioning

confidence: 99%

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Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Kitaoka¹

2017

Uniform Distribution Theory

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ABSTRACT. Let f (x) be a monic polynomial in Z[x] with roots α 1 , . . . , α n . We point out the importance of linear relations among 1, α 1 , . . . , α n over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers. Communicated by Shigeki AkiyamaIn this note, a polynomial means always a monic one over the ring Z of integers and the letter p denotes a prime number, unless specified. Letbe a polynomial of degree n. As in the previous papers, we putis fully splitting modulo p} for a positive number X and Spl (f ) := Spl ∞ (f ). In this note, we require the following conditions on the local roots r 1 , . . . , r n (∈ Z) of f (x) ≡ 0 mod p for a prime p ∈ Spl (f ) :We can determine local roots r i uniquely with this global ordering. If f is irreducible and of deg(f ) > 1, and p is sufficiently large, then (2) is equivalent to 0 < r 1 < · · · < r n < p. Here, we consider two types of distribution of local roots r i of f .

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Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on 'elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on 'Artin for K-theory of number fields', and Hester Graves (together with me) on 'Artin's conjecture and Euclidean domains'.

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Densities of Sets of Primes Related to Decimal Expansion of Rational Numbers

Cited by 5 publications

References 2 publications

A statistical relation of roots of a polynomial in different local fields

A statistical relation of roots of a polynomial in different local fields

Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Artin's Primitive Root Conjecture – A Survey

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