Related papers: Large primes in generalized Pascal triangles
The purpose of this study is to show how to get a necessary criterion for prime numbers with the help of special matrices. My special interest lies in the empirical research of these matrices and their patterns, structures and symmetries.…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
In this paper, we prove a necessary and sufficient condition for the Lucas-Carmichael integers in terms of the sum of base-$p$ digits. We also study some interesting properties of such integers. Finally, we prove that there are infinitely…
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in…
The purpose of this article is to delve into the properties of invariants. The properties, explained in [2], reveal new ways to develop algorithms that allow us to test the primality of a number. In this article, some of these are shown,…
We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…
We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
For $a \neq 1$ and $p$ prime, we define numbers of the form $pa^2$ to be Square-Prime (SP) Numbers. For example, 75 = 3 $\cdot$ 25; 108 = 3 $\cdot$ 36; 45 = 5 $\cdot$ 9. These numbers are listed in the OEIS as A228056. We study the…
We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…
The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to…
Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving…
We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows…
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…
This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
In this paper, we provide a generalization of Proth's theorem for integers of the form $Kp^n+1$. In particular, a primality test that requires only one modular exponentiation similar to that of Fermat's test without the computation of any…
In the classic "Concrete Math", by Graham, Patashnik and Knuth, it is stated that "The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it is not too surprising that we can find some surprising…
We found a regularity of the behavior of primes that allows to represent both prime and natural numbers as infinite matrices with a common formation rule of their rows. This regularity determines a new class of infinite cyclic groups that…