Displaying similar documents to “Small points on a multiplicative group and class number problem”

Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves

Joseph H. Silverman (2012)

Journal de Théorie des Nombres de Bordeaux

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A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to  1 modulo  m . We prove a similar result for polynomials  f ( X ) that are divisible in  ( ℤ / m ℤ ) [ X ] by a polynomial of the form 1 + X + ⋯ + X n for some n ≥ ϵ deg ( f ) . We also formulate and prove an analogous statement for elliptic curves.

Smooth solutions to the a b c equation: the x y z Conjecture

Jeffrey C. Lagarias, Kannan Soundararajan (2011)

Journal de Théorie des Nombres de Bordeaux

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This paper studies integer solutions to the a b c equation A + B + C = 0 in which none of A , B , C have a large prime factor. We set H ( A , B , C ) = max ( | A | , | B | , | C | ) , and consider primitive solutions ( gcd ( A , B , C ) = 1 ) having no prime factor larger than ( log H ( A , B , C ) ) κ , for a given finite κ . We show that the a b c Conjecture implies that for any fixed κ < 1 the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed κ > 8 the a b c equation has infinitely many primitive...

On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

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For every positive integer n let p ( n ) be the largest prime number p ≤ n . Given a positive integer n = n 1 , we study the positive integer r = R ( n ) such that if we define recursively n i + 1 = n i - p ( n i ) for i ≥ 1 , then n r is a prime or 1 . We obtain upper bounds for R ( n ) as well as an estimate for the set of n whose R ( n ) takes on a fixed value k .

Small generators of function fields

Martin Widmer (2010)

Journal de Théorie des Nombres de Bordeaux

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Let 𝕂 / k be a finite extension of a global field. Such an extension can be generated over k by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.

A note on the diophantine equation k 2 - 1 = q n + 1

Maohua Le (1998)

Colloquium Mathematicae

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In this note we prove that the equation k 2 - 1 = q n + 1 , q ≥ 2 , n ≥ 3 , has only finitely many positive integer solutions ( k , q , n ) . Moreover, all solutions ( k , q , n ) satisfy k 10 10 182 , q 10 10 165 and n 2 · 10 17 .