Prime Research Articles

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions Ff;N,pm=∑k≤Nfpmk, m∈Z+∪{0} associated with the multiplicative arithmetic functions f of a scaled variable k∈Z+, where p is a prime number. Assuming an asymptotic behavior of the summatory function, F{f;N,1}=N→∞G1(N)1+OG2(N), where G1(N)=Na1logNb1, G2(N)=N−a2logN−b2 and a1,a2≥0, −∞<b1,b2<∞, we calculate the renormalization function Rf;N,pm, defined as a ratio Ff;N,pm/F{f;N,1}, and find its asymptotics R∞f;pm when N→∞. We prove that a renormalization function is multiplicative, i.e., R∞f;∏i=1npimi=∏i=1nR∞f;pimi with n distinct primes pi. We extend these results to the other summatory functions ∑k≤Nf(pmkl), m,l,k∈Z+ and ∑k≤N∏i=1nfikpmi, fi≠fj, mi≠mj. We apply the derived formulas to a large number of basic summatory functions including the Euler ϕ(k) and Dedekind ψ(k) totient functions, divisor σn(k) and prime divisor β(k) functions, the Ramanujan sum Cq(n) and Ramanujan τ Dirichlet series, and others.

Primes as Sums of Fibonacci Numbers

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 k there exists a prime number that can be represented as the sum of k k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. In our case, these morphic sequences are based on the Zeckendorf expansion of a positive integer n n — we write n n as the sum of non-consecutive Fibonacci numbers. More precisely, we are concerned with the Zeckendorf sum-of-digits function z \mathsf {z} , which returns the minimal number of Fibonacci numbers needed to write a positive integer as their sum. The proof of such a prime number theorem for z \mathsf {z} , combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. Problems of this type have been discussed intensively in the context of the base- q q expansion of integers. The driving forces of this development were the Gelfond problems (1968/1969), more specifically the behavior of the sum-of-digits function in base q q along the sequence of primes and along integer-valued polynomials, and the Sarnak conjecture. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009). Later the second author (2017) proved Sarnak’s conjecture for the class of automatic sequences, which are based on the q q -ary expansion of integers, and which generalize the sum-of-digits function in base q q considerably. For the (partial) solution of the Gelfond problems (1967/1968), Mauduit and Rivat have developed a powerful method that is based on techniques for “cutting off digits”, on sophisticated estimates for Fourier terms, and on estimates for exponential sums. These techniques — together with a new decomposition of finite automata — were also the basis for the second author’s result on automatic sequences. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat’s method considerably. In fact, we are departing significantly from this method, proving the statement that exp ⁡ ( 2 π i ϑ z ( n ) ) \exp (2\pi i \vartheta \hspace {0.5pt}\mathsf {z}(n)) has level of distribution 1 1 . This latter result forms an essential part of our treatment of the occurring sums of type I \mathrm {I} and I I \mathrm {II} and uses Gowers norms related to the Zeckendorf sum-of-digits function as a central technical tool. Gowers norms are a higher order generalization of the above-mentioned Fourier terms, and their appearance in our method is intimately tied to the iterated application of a new generalization of van der Corput’s inequality.

Prospects for the Use of Quasi-Mersen Numbers in the Design of Parallel-Serial Processors

It is shown that a serial-parallel processor, comparable in bit capacity to a 16-bit binary processor, can be implemented based on an algorithm built on the residue number system, a distinctive feature of which is the use of the first four quasi-Mersenne numbers, i.e., prime numbers representable as pk=2k+1, k=1,2,3,4. Such a set of prime numbers satisfies the criterion 2p1p2p3p4+1=P, where P is also a prime number. Fulfillment of this criterion ensures the possibility of convenient use of the considered RNS for calculating partial convolutions developed for the convenience of using convolutional neural networks. It is shown that the processor of the proposed type can be based on the use of a set of adders modulo a quasi-Mersenne number, each of which operates independently. A circuit of a modulo 2k+1 adder is proposed, which can be called a trigger circuit, since its peculiarity is the existence (at certain values of the summed quantities) of two stable states. The advantage of such a circuit, compared to known analogs, is the simplicity of the design. Possibilities for further development of the proposed approach related to the use of the digital logarithm operation, which allows reducing the operations of multiplication modulo 2k+1 to addition operations, are discussed.

Lower bounds for the number of number fields with Galois group GL2(𝔽ℓ)

Abstract Let ℓ ≥ 5 {\ell\geq 5} be a prime number and let 𝔽 ℓ {\mathbb{F}_{\ell}} denote the finite field with ℓ {\ell} elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ⁡ ( 𝔽 ℓ ) {\operatorname{GL}_{2}(\mathbb{F}_{\ell})} and absolute discriminant bounded above by X is asymptotically at least X ℓ / ( 12 ⁢ ( ℓ - 1 ) ⁢ # ⁢ GL 2 ⁡ ( 𝔽 ℓ ) ) log ⁡ X {\frac{X^{{\ell}/({12(\ell-1)\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})})}}{% \log X}} . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ⁡ ( ℚ ¯ / ℚ ) → GL 2 ⁡ ( 𝔽 ℓ ) {\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{% GL}_{2}(\mathbb{F}_{\ell})} ordered by the prime to ℓ {\ell} part of the Artin conductor of ρ.

Matrix-based homomorphic encryption-using random prime numbers

Matrix-based homomorphic encryption-using random prime numbers

SEIDEL LAPLACIAN ENERGY of ZERO-DIVISOR GRAPH Г[Z_n

The Seidel Laplacian energy of graphs has recently been defined. In the present work, we compute the Seidel Laplacian energy of the zero-divisor graph Г[Z_n] for n=p^2,n=pq, n=2q, and n=p^3, where p,q are distinct prime numbers.

Finite Local Rings of Length 4

This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field Fpm, where p is a prime number. Such rings have an order of p4m elements. The current paper provides the structure and classification, up to isomorphism, of local rings consisting of p4m elements. We also give the exact number of non-isomorphic classes of these rings with fixed invariants p,n,m,k. In particular, we have listed all finite local rings of 4-length and of order p8 and 256.

P-Nilpotent Maximal Subgroups in Finite Groups

Let p be a prime number and suppose that every maximal subgroup of a finite group is either p-nilpotent or has prime index. Such a group need not be p-solvable, and we study its structure by proving that only one nonabelian simple group of order divisible by p, which belongs to the family PSLn(q), can be involved in it. For p=2, we specify more, and in fact, such a simple group must be isomorphic to PSL2(ra) for certain values of the prime r and the parameter a.

Computing Riemann zeros with light scattering

Finding hidden order within disorder is a common interest in material science, wave physics, and mathematics. The Riemann hypothesis, stating the locations of nontrivial zeros of the Riemann zeta function, tentatively characterizes statistical order in the seemingly random distribution of prime numbers. This famous conjecture has inspired various connections with different branches of physics, recently with non-Hermitian physics, quantum field theory, trapped-ion qubits, and hyperuniformity. Here we develop the computing platform for the Riemann zeta function by employing classical scattering of light. We show that the Riemann hypothesis suggests the landscape of semi-infinite optical scatterers for the perfect reflectionless condition under the Born approximation. To examine the validity of the scattering-based computation, we investigate the asymptotic behaviors of suppressed reflections with the increasing number of scatterers and the emergence of multiple scattering. The result provides another bridge between classical physics and the Riemann zeros, exhibiting the design of wave devices inspired by number theory. Published by the American Physical Society 2024

Special Pair of Rectangles and Wagstaff Primes

The objective of this work is to identify the pairs of rectangles such that a Wagstaff prime number represents the sum of their areas in each pair. Each Wagstaff prime number is also accompanied by the number of primitive and non-primitive rectangles. Key Words: Pair of rectangles, Wagstaff prime number, Primitive, Non-primitive, Area.

Periodic Groups Saturated with Finite Frobenius Groups with Complements of Orders Divisible by a Prime Number

Periodic Groups Saturated with Finite Frobenius Groups with Complements of Orders Divisible by a Prime Number

Commutative Chain Rings with Index of Nilpotency 5 and Residue Field Fpm

This paper gives a thorough characterization of chain rings with index of nilpotency 5 and residue field Fpm, where p represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing p5m elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts.

Alternating state complexity of the set of primes and squarefree integers

AbstractWe show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.

Herr complex of (φ,τ)-modules

Let p be an odd prime number and K a mixed characteristic complete discrete valuation field with perfect residue field of characteristic p. We construct a three-term complex, defined in terms of the (φ,τ)-module of a p-adic representation and prove its homology is isomorphic to the Galois cohomology of the representation. We further show that our complex is quasi-isomorphic to the four-term complex constructed by Tavares Ribeiro, providing an alternative proof of our result. As an application, we describe Galois cohomology of the Tate module associated to a p-divisible group in terms of corresponding Breuil-Kisin modules.

CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES

Abstract For a prime p and a rational elliptic curve $E_{/\mathbb {Q}}$ , set $K=\mathbb {Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$ . The class group $\operatorname {Cl}_K$ is a module over $\operatorname {Gal}(K/\mathbb {Q})$ . Given a fixed odd prime number p, we study the average nonvanishing of certain Galois stable quotients of the mod-p class group $\operatorname {Cl}_K/p\operatorname {Cl}_K$ . Here, E varies over all rational elliptic curves, ordered according to height. Our results are conditional, since we assume that the p-primary part of the Tate–Shafarevich group is finite. Furthermore, we assume predictions made by Delaunay for the statistical variation of the p-primary parts of Tate–Shafarevich groups. We also prove results in the case when the elliptic curve $E_{/\mathbb {Q}}$ is fixed and the prime p is allowed to vary.

Improving the RSA Encryption for Images by Introducing DNA Sequence Encoding

Recent research is focused on the exploitation of DNA-based molecules for data encryption due to their high capacity to store larger volumes of data and lower computation requirements [1, 2]. This study proposes a Hybrid Image Encryption method (HIE) that convolves DNA sequence encoding with the Rivest–Shamir–Adleman (RSA) algorithm to enhance the security of image encryption. The proposed scheme uses small prime numbers to encrypt the image, which is then encoded as a DNA sequence. Subsequently, the encrypted DNA sequence is stored in a physical medium. The encrypted DNA sequence can then be decrypted using the RSA algorithm and the corresponding private key to recover the original image. The results show that using small prime numbers for RSA encryption of an image and encoding it as a DNA sequence can enhance security and reduce computational time.

Some new results on the largest cycle consisting of quadratic residues

The length of the largest cycle consisting of quadratic residues of a positive integer n is denoted by L(n). In this paper, we have obtained a formula for finding L(p), where p is a prime. Also, we attempt to characterize a prime number p in terms of the largest cycle consisting of quadratic residues of p.

On the Binary Goldbach Conjecture: Analysis and Alternate Formulations Using Projection, Optimization, Hybrid Factorization, Prime Symmetry and Analytic Approximation

An analysis, based on different mathematical approaches, of the binary Goldbach conjecture −which states that every even integer s≥6 is the sum of two odd primes, called Goldbach primes− is presented. Each approach leads to a different reformulation of this conjecture, thus contributing unique insights into the structure, properties and distribution of prime numbers. The above-mentioned reformulations are based on the following distinct, interrelated and complementary approaches: projection, optimization, hybrid prime factorization, prime symmetry and analytic approximation. Additionally, it is shown that prime factorization is an optimal projection operation on the set of integers; that Goldbach pairs correspond to solutions of an optimization problem; that hybrid prime factorization can be used to generate Goldbach primes; that prime symmetry, a powerful property of Goldbach primes, can be used to validate the binary Goldbach conjecture in short intervals, and to determine the rules that govern the “algebraic evolution” of Goldbach pairs, as the value of s increases; and that analytic approximation, using translational and rotational shifts of smooth functions, leads to a useful approximation of a primality test function and the prime counting function π(s). The paper’s findings support the broader hypothesis that prime numbers, by virtue of their optimality in representing, additively and multiplicatively, any measurable quantity in the universe, supported by the Fundamental Theorem of Arithmetic and the binary Goldbach conjecture, may be a viable alternative to the exclusive use of binary logic, as a means of achieving additional computational efficiencies of scale in the future.

Vieta’s Formula, the Fabius Function and the Partition Function

We use an idea of Pólya and Szegö to give a common basis to Vieta’s formula, Fabius function and the partition function. Moreover our construction leads also to a function considered by Hallström, Bowen and Macintyre, which has, as a particular value, the Kepler-Bouwkamp constant, and to a function considered by Zondadari, that vanishes only at prime numbers.

The average number of Goldbach representations and Zero-free regions of the Riemann zeta function

In this paper, we prove an unconditional form of Fujii’s formula for the average number of Goldbach representations and show that the error in this formula is determined by a general zero-free region of the Riemann zeta function, and vice versa. In particular, we describe the error in the unconditional formula in terms of the remainder in the Prime Number Theorem which connects the error to zero-free regions of the Riemann zeta function.