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Nombres de racines d’un polynôme entier modulo q

Monique Branton, Olivier Ramaré (1998)

Journal de théorie des nombres de Bordeaux

Nous montrons que l’ensemble des racines modulo une puissance d’un nombre premier d’un polynôme à coefficients entiers de degré d est une union d’au plus d progressions arithmétiques de modules assez grands. Nous en déduisons une majoration du nombre de ses racines dans un intervalle réel court.

Nonaliquots and Robbins numbers

William D. Banks, Florian Luca (2005)

Colloquium Mathematicae

Let φ(·) and σ(·) denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of m ≤ x for which the equation m = σ(n) - n has no solution. We also show that the set of positive integers m not of the form (p-1)/2 - φ(p-1) for some prime number p has a positive lower asymptotic density.

Non-Wieferich primes in number fields and a b c -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

Let K / ℚ be an algebraic number field of class number one and let 𝒪 K be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in 𝒪 K under the assumption of the a b c -conjecture for number fields.

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

Let Q = ( q n ) n = 1 ∞ be a sequence of bases with q i ≥ 2 . In the case when the q i are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q -Cantor series expansion is both Q -normal and Q -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q , and from this construction we can provide computable constructions of numbers with atypical normality properties.

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Kevin J. McGown (2012)

Journal de Théorie des Nombres de Bordeaux

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant Δ = 7 2 , 9 2 , 13 2 , 19 2 , 31 2 , 37 2 , 43 2 , 61 2 , 67 2 , 103 2 , 109 2 , 127 2 , 157 2 . A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree ℓ and conductor f . Assume the GRH for ζ K ( s ) . If 38 ( ℓ - 1 ) 2 ( log f ) 6 log log f < f , then K is not norm-Euclidean.

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