Displaying similar documents to “Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials”

Heights of roots of polynomials with odd coefficients

J. Garza, M. I. M. Ishak, M. J. Mossinghoff, C. G. Pinner, B. Wiles (2010)

Journal de Théorie des Nombres de Bordeaux

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Let α be a zero of a polynomial of degree n with odd coefficients, with α not a root of unity. We show that the height of α satisfies h ( α ) 0 . 4278 n + 1 . More generally, we obtain bounds when the coefficients are all congruent to 1 modulo m for some m 2 .

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 .

Landau’s function for one million billions

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

Journal de Théorie des Nombres de Bordeaux

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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 ( 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 ( 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...

On the mean square of the divisor function in short intervals

Aleksandar Ivić (2009)

Journal de Théorie des Nombres de Bordeaux

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We provide upper bounds for the mean square integral X 2 X 𝔻 k ( x + h ) - 𝔻 k ( x ) 2 d x , where h = h ( X ) 1 , h = o ( x ) as X and h lies in a suitable range. For k 2 a fixed integer, 𝔻 k ( x ) is the error term in the asymptotic formula for the summatory function of the divisor function d k ( n ) , generated by ζ k ( s ) .

On the counting function for the generalized Niven numbers

Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño (2009)

Journal de Théorie des Nombres de Bordeaux

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Given an integer base q 2 and a completely q -additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function N f ( x ) = # 0 n < x | f ( n ) n under a mild restriction on the values of f . When f = s q , the base q sum of digits function, the integers counted by N f are the so-called base q Niven numbers, and our result provides a generalization of the asymptotic known in that case.

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

Yasutsugu Fujita, Nobuhiro Terai (2011)

Journal de Théorie des Nombres de Bordeaux

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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 [ k , l ] .