Related papers: Elementary Primes Counting Methods
B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any…
We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely…
Let $P$ be a subset of the primes of lower density strictly larger than $\frac12$. Then, every sufficiently large even integer is a sum of four primes from the set $P$. We establish similar results for $k$-summands, with $k\geq 4$, and for…
We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less…
The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
Let $K\geq 2$ be a natural number and $a_i,b_i\in\mathbb{Z}$ for $i=1,\ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $p\leq x$ for which all $a_ip+b_i$…
We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…
A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The…
Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to…
This monograph considers a few topics in the theory of primitive roots g(p) modulo a prime p>=2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g^*(p) modulo p, a large prime, are determined. One of…
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation…
The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…
For integers x and k, let T(x;2k) denote the number of twin prime pairs (p,p+2k) with a distance 2k<=2x**0.5 and p<=x (not p+2k<=x). Let Tg(x;2x**0.5) denote the average of T(x;2k) for all 2k<=2x**0.5. Logically, T(x;2k) should be a…
A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.
We characterize models of Peano arithmetic (PA) with infinitely many infinite primes p such that p + 2 has no finite prime divisor.
Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…
Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…