相关论文: On the Emergence of the Quanta Prime Sequence

This paper highlights the emergence of the Omega sequence in number theory and its connection with the emergence of a new prime number, and also highlights its theoretical applications for Lucas-Lehmer primality test, and Euclid-Euler…

We prove new theorems for the polynomial expansions of $x^n \pm y^n$ in terms of the binary quadratic forms $\alpha x^2 + \beta xy + \alpha y^2 $ and $a x^2 + bxy + a y^2 $. The paper gives new arithmetic differential approach to compute…

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the…

In this paper we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev…

We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x…

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$)…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers.…

The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as…

Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between…

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set…

A simple criterion is derived in order that a number sequence ${\cal S}_n$ is a permitted spectrum of a quantized system. The sequence of the prime numbers fulfils the criterion and the corresponding one-dimensional quantum potential is…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

Recent oracle separations [Kretschmer, TQC'21, Kretschmer et. al., STOC'23] have raised the tantalizing possibility of building quantum cryptography from sources of hardness that persist even if the polynomial hierarchy collapses. We…

This article is meant to provide an additional point of view, applying known knowledge, to supply keys that have a series of non-repeating digits, in a manner that is not usually thought of. Traditionally, prime numbers are used in…

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…

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.…

Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…

Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy. Developing new algorithms or improving upon existing ones could…