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# The exceptional set for the distribution of primes between consecutive powers

Author: D. Bazzanella

## Abstract

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the afore-mentioned conjecture.

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# Small gaps between primes

Author: J. Sivak-Fischler

## Abstract

Combining Goldston-Yildirim’s method on k-correlations of the truncated von Mangoldt function with Maier’s matrix method, we show that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\Xi _r : = \lim \inf _{n \to \infty } \tfrac{{p_{n + r} - p_n }} {{\log p_n }} \leqq e^{ - \gamma } \left( {r - \tfrac{{\sqrt r }} {2}} \right)$$ \end{document}
for all r ≧ 1 where p n denotes the nth prime number and γ is Euler’s constant. This is the best known result for any r ≧ 11.

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# Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

Authors: Milad Ahanjideh and Neda Ahanjideh

Let V be the 2-dimensional column vector space over a finite field

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}
(where q is necessarily a power of a prime number) and let ℙq be the projective line over
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}
. In this paper, it is shown that GL 2(q), for q ≠ 3, and SL 2(q) acting on V − {0} have the strict EKR property and GL 2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GL n(q) acting on
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document}
has the EKR property and the derangement graph of PSL 2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

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# On the distribition of residue classes of quadratic forms and integer-detecting sequences in number fields

Authors: C. Elsner and J. W. Sander

MOTOHASHI, Y., Lectures on sieve methods and prime number theory , Tata Institute of Fundamental Research, Lectures on Mathematics and Physics, 72 , Published for the Tata Institute of Fundamental Research

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# The least common multiple of a sequence of products of linear polynomials

Authors: Shaofang Hong, Guoyou Qian and Qianrong Tan

Amer. Math. Monthly 89 126 – 129 10.2307/2320934 . [10] Selberg , A. 1950 An elementary proof of the prime-number theorem for

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# A remark on a theorem of the Goldbach-Waring type

Author: C. Bauer

generalized Vaughan’s identity , Canadian J. Math. 34 ( 1982 ), 1365 – 1377 . MR 84q:10075 [5] H ua , L. K. , Some results in the additive prime number theory

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# An L-function free proof of Hua's Theorem on sums of five prime squares

Author: Claus Bauer

in the additive prime number theory , Quart. J. Math. , 9 ( 1938 ), 68 – 80 . [8] L i , H. and P an , H. , A density version of Vinogradov's three primes theorem

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# Modular constructions of pseudorandom binary sequences with composite moduli

Authors: Joël Rivat and András Sárközy

## Summary

Recently, Goubin, Mauduit, Rivat and Srkzy have given three constructions for large families of binary sequences. In each of these constructions the sequence is defined by modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} congruences where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} is a prime number. In this paper the three constructions are extended to the case when the modulus is of the form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $q$ \end{document} are two distinct primes not far apart (note that the well-known Blum-Blum-Shub and RSA constructions for pseudorandom sequences are also of this type). It is shown that these modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions also have certain strong pseudorandom properties but, e.g., the (long range'') correlation of order \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $4$ \end{document} is large (similar phenomenon may occur in other modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions as well).

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# On the sum of a prime and a k-th power of prime in short intervals

Author: Y. C. Wang

] Hua , L. K. 1938 Some results in prime number theory Q.J. Math. 9 68 – 80 10.1093/qmath/os-9.1.68 . [5

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# The values of additive forms at prime arguments

Author: Haiwei Sun

. Soc. 2006 140 1 13 Hua, L. K. , Some results in the additive prime number

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