Do you want to publish a course? Click here

Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

136   0   0.0 ( 0 )
 Added by Liangyi Zhao
 Publication date 2014
  fields
and research's language is English




Ask ChatGPT about the research

Let $t in mathbb{N}$, $eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q leq x^{5/12-eta}$, $q$ not a multiple of the conductor of the exceptional character $chi^*$ (if it exists). Suppose further that, [ max {p : p | q } < exp (frac{log x}{C log log x}) ; ; {and} ; ; prod_{p | q} p < x^{delta}, ] where $C$ and $delta$ are suitable positive constants depending on $t$ and $eta$. Let $a in mathbb{Z}$, $(a,q)=1$ and [ mathcal{A} = {n in (x/2, x]: n equiv a pmod{q} } . ] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $mathcal{A}$ with [ p_t - p_1 ll qt exp (frac{40 t}{9-20 theta}) . ] Here $theta = (log q) / log x$.



rate research

Read More

145 - Lynn Chua , Soohyun Park , 2014
We use Maynards methods to show that there are bounded gaps between primes in the sequence ${lfloor nalpharfloor}$, where $alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $lfloor f(q)rfloor$ with $q$ a positive integer.
106 - Chunlei Liu 2021
It is proven that there are infinitely prime pairs whose difference is no greater than 20.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 le r le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea of our approach is summarized on the creation and on the analyzing sequence of sets of distinct co-primes with the first $n$ primes, $left{ p_i :, ileq n right}$, and the important properties of the modulus linear combination of the co-prime sets, $H=left(1,p_{n+1},..., Pi_{i=1}^n p_i-1right) $, that gives sets of even numbers ${0,2,4,..., Pi_{i=1}^n p_i -2 }$. Furthermore, by generalizing our approach, the Polignac conjecture the existence of infinitely many cousin primes, $p_{n+1}-p_{n}=4$, and the statement that every even integer can be expressed as a difference of two primes, are derived as well.
171 - Yuanyou Furui Cheng 2013
We give an explicit form of Inghams Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $xsp{3}$ and $(x+1)sp{3}$ if $loglog xge 15$.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا