A note on primes in certain residue classes

2019

Let p be a prime number. In [1], Booker and Pomerance find an integer y with 1 < y ≤ p such that all non-zero residue classes modulo p can be written as a square-free product of positive integers up to y. Let us denote by y(p) the smallest such y. Booker and Pomerance show in their paper that except for p = 5 and 7, we have y(p) ≤ y and some better upper bounds were conjectured. Later, Munsch and Shparlinski [7] proved those conjectures with even better localization. Their work was done as the same time as ours, but with fairly more complicated methods in the proof. We were seeking to find a solution for the problem using Pólya-Vinogradov inequality or at most its improvement, the Burgess bound on character sums. That being said, we removed the condition in the problem that the product has to be square-free. We proved that for m > p √ , each residue class b of (Z/pZ)× can be written as a product of elements of the set {1, 2, . . . ,m} modulo p. In fact, we showed that the numb...

2013

In this paper we discuss Legendre’s, Brocard’s, Andrica’s, and Oppermann’s conjectures. We propose a conjecture regarding the distribution of prime numbers and we also prove that if it is true, then the previously mentioned conjectures follow. Moreover, we also show that if the conjecture in question holds, then there is at least one prime number in the interval [n, n+ 2 b √ nc − 1] for every positive integer n.

Journal of the London Mathematical Society, 1993

arXiv (Cornell University), 2018

Proceedings - Mathematical Sciences, 2013

Given a prime number l, a finite set of integers S = {a1,. .. , am} and m many l th roots of unity ζ r i l , i = 1,. .. , m we study the distribution of primes p in Q(ζ l) such that the l th residue symbol of ai with respect to p is ζ r i l , for all i. We find out that this is related to the degree of the extension Q(a 1 l 1 ,. .. , a 1 l m)/Q. We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S = {a1,. .. , am}.

2021

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k + 1, receives attention.

The Quarterly Journal of Mathematics, 1986

arXiv: Number Theory, 2018

We estimate the asymptotic density of the set $\\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\\bar{B} \\subset \\bar{A}$ such that for $p_1, p_2 \\in \\bar{B}$ no common prime divisor of $p_1(p_1 + 1)(p_1 - 1)$ and $p_2 (p_2 + 1)(p_2 - 1)$ is larger than $3$. Assuming a generalized Hardy--Littlewood conjecture, we prove that for both $\\bar{A}$ and $\\bar{B}$ the number of elements lesser than $x$ is asymptotically equal to a constant times $ x / (\\log x)^3$.

2008

A set A of positive integers is relatively prime to n if gcd(A∪{n}) = 1. Given positive integers l ≤ m ≤ n, let Φ([l,m], n) denote the number of nonempty subsets of {l, l +1, . . . ,m} which are relatively prime to n and let Φk([l,m], n) denote the number of such subsets of cardinality k. In this paper we give formulas for these functions for the case l = 1. Intermediate consequences include identities for the number of subsets of {1, 2, . . . , n} with elements in both {1, 2, . . . ,m} and {m,m + 1, . . . , n} which are relatively prime to n and the number of such subsets having cardinality k. Some of our proofs use the Möbius inversion formula extended to functions of several variables.

2017

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cram\\'{e}r's, Shanks', and five Smarandache's conjectures. We make use of both Firoozbakht's conjecture (which recently proved by the author) and Kourbatov's theorem on the distribution of and gaps between consecutive primes. These latter conjecture and theorem play an essential role in our methods for proving these famous conjectures. In order to prove Shanks' conjecture, we make use of Panaitopol's asymptotic formula for $\\pi(x)$ as well.

Mathematics of Computation, 2004

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 10 20 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 10 18 .

Acta Arithmetica, 2007

Mathematische Zeitschrift, 1996

Inventiones Mathematicae, 1984