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On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

In this paper we consider the asymptotic formula for the number of the solutions of the equation p 1 + p 2 + p 3 = N where N is an odd integer and the unknowns p i are prime numbers of the form p i = [ n 1 / γ i ] . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case γ 1 = γ 2 = γ 3 = γ our theorem implies that every sufficiently large odd integer N may be written as the sum of three Piatetski-Shapiro primes of type γ for 50 / 53 < γ < 1 .

On the positivity of the number of t-core partitions

Ken Ono (1994)

Acta Arithmetica

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n . Here we define a special class of partitions. 1. Let t 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof n . The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3, 4, 6]. If t 1 and n 0 , then we define...

On the powerful part of n 2 + 1

Jan-Christoph Puchta (2003)

Archivum Mathematicum

We show that n 2 + 1 is powerfull for O ( x 2 / 5 + ϵ ) integers n x at most, thus answering a question of P. Ribenboim.

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