Related papers: Probabilistic method of proving twin primes' infin…
We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…
The twin prime conjecture asserts that there are infinitely many pairs of primes that differ by two. While recent advances have improved our understanding of bounded prime gaps, the conjecture remains unresolved. This paper refines the…
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more…
A dynamic sieve method is designed according to the basic sieve method. It mainly refers to the symbolic dynamics theory. By this method, we could connect the prime system with familiar 'Logistic Mapping'. An interesting discovery is that…
This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…
We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…
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.
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic…
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
We resolve the function field analogue of the conjecture concerning distribution of twin primes in arithmetic progression.
We give an account of the arguments that lead from the assumption of the existence of exceptional characters to the asymptotics in related ranges for the counting function of twin primes.
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.
We adopt a physically motivated empirical approach to the characterisation of the distributions of twin and triplet primes within the set of primes, rather than in the set of all natural numbers. Remarkably, the occurrences of twins or…
This paper is based on a talk given to motivated high school (and younger) students at a BAMA (Bay Area Math Adventure) event. Some of the methods used to study primes and twin primes are introduced.
We study a special set of constellations of primes generated by twin primes.
In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.