Displaying similar documents to “Power-free values, large deviations, and integer points on irrational curves”

Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner, Takao Komatsu, Iekata Shiokawa (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We compute upper and lower bounds for the approximation of hyperbolic functions at points 1 / s ( s = 1 , 2 , ⋯ ) by rationals x / y , such that x , y satisfy a quadratic equation. For instance, all positive integers x , y with y ≡ 0 ( mod 2 ) solving the Pythagorean equation x 2 + y 2 = z 2 satisfy | y sinh ( 1 / s ) - x | ≫ log log y log y . Conversely, for every s = 1 , 2 , ⋯ there are infinitely many coprime integers x , y , such that | y sinh ( 1 / s ) - x | ≪ log log y log y and x 2 + y 2 = z 2 hold simultaneously for some integer z . A generalization to the approximation of h ( e 1 / s ) for rational...

Landau’s function for one million billions

Marc Deléglise, Jean-Louis Nicolas, Paul Zimmermann (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let 𝔖 n denote the symmetric group with n letters, and g ( n ) the maximal order of an element of 𝔖 n . If the standard factorization of M into primes is M = q 1 α 1 q 2 α 2 ... q k α k , we define ℓ ( M ) to be q 1 α 1 + q 2 α 2 + ... + q k α k ; one century ago, E. Landau proved that g ( n ) = max ℓ ( M ) ≤ n M and that, when n goes to infinity, log g ( n ) ∼ n log ( n ) . There exists a basic algorithm to compute g ( n ) for 1 ≤ n ≤ N ; its running time is 𝒪 N 3 / 2 / log N and the needed memory is 𝒪 ( N ) ; it allows computing g ( n ) up to, say, one million. We describe an algorithm to calculate g ( n ) for n up to 10 15 . The main idea is to use the...

The divisor problem for binary cubic forms

Tim Browning (2011)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over ℚ and discuss some applications.

Generators for the elliptic curve y 2 = x 3 - n x

Yasutsugu Fujita, Nobuhiro Terai (2011)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let E be an elliptic curve given by y 2 = x 3 - n x with a positive integer n . Duquesne in 2007 showed that if n = ( 2 k 2 - 2 k + 1 ) ( 18 k 2 + 30 k + 17 ) is square-free with an integer k , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of E . In this paper, we generalize this result and show that the same is true for infinitely many binary forms n = n ( k , l ) in ℤ [ k , l ] .