scholarly journals Prime numbers of the form p=m2+n2+1 in short intervals

2007 ◽  
Vol 128 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Kaisa Matomäki
Keyword(s):  
Prime Numbers ◽  
Short Intervals

On prime numbers of special kind on short intervals

2006 ◽  
Vol 79 (5-6) ◽  
pp. 848-853
Author(s):  
N. N. Mot’kina
Keyword(s):  
Special Kind ◽  
Prime Numbers ◽  
Short Intervals

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

Sums of Two Squares in Short Intervals

2000 ◽  
Vol 52 (4) ◽  
pp. 673-694 ◽  
Author(s):  
Antal Balog ◽  
Trevor D. Wooley
Keyword(s):  
Prime Numbers ◽  
Positive Dimension ◽  
Short Intervals ◽  
Sums Of Two Squares ◽  
Sieving Process

AbstractLet denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of than expected, and infinitely many intervals containing considerably fewer than expected.

scholarly journals Primes in explicit short intervals on RH

2016 ◽  
Vol 12 (05) ◽  
pp. 1391-1407 ◽  
Author(s):  
Adrian W. Dudek ◽  
Loïc Grenié ◽  
Giuseppe Molteni
Keyword(s):  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Upper Bounds ◽  
Short Intervals ◽  
The Difference

In this paper, on the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

scholarly journals Prime numbers in short intervals

1985 ◽  
Vol 17 (1) ◽  
pp. 419-429
Author(s):  
J. Pintz
Keyword(s):  
Prime Numbers ◽  
Short Intervals

Exceptional characters and prime numbers in short intervals

2004 ◽  
Vol 10 (1) ◽  
pp. 61-69 ◽  
Author(s):  
John B. Friedlander ◽  
Henryk Iwaniec
Keyword(s):  
Prime Numbers ◽  
Short Intervals

scholarly journals SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

2019 ◽  
Vol 109 (3) ◽  
pp. 351-370 ◽  
Author(s):  
ALESSANDRO LANGUASCO ◽  
ALESSANDRO ZACCAGNINI
Keyword(s):  
Asymptotic Formula ◽  
Prime Numbers ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Representations Of Integers ◽  
Binary Problems

AbstractWe improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

scholarly journals Waring–Goldbach Problem of Even Powers in Short Intervals

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Liqun Hu ◽  
Tanhui Zhang
Keyword(s):  
Prime Numbers ◽  
Short Intervals ◽  
Goldbach Problem ◽  
Average Behaviour

In this paper, we study the average behaviour of the representations of n = p 1 2 + p 2 4 + p 3 4 + p 4 k over short intervals for k ≥ 4 , where p 1 , p 2 , p 3 , p 4 are prime numbers. This improves the previous results.

scholarly journals Almost all short intervals containing prime numbers

1996 ◽  
Vol 76 (1) ◽  
pp. 21-84 ◽  
Author(s):  
Chaohua Jia
Keyword(s):  
Prime Numbers ◽  
Short Intervals ◽  
Almost All
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