Prime Research Articles

Corrigendum to ‘Explicit interval estimates for prime numbers’

This article corrects a mistake in ‘Explicit interval estimates for prime numbers’, Math. Comp. 91 (2022), 1955–1970. The error was in a closed-form expression for an integral involving the smoothing function. The table of pairs of ( Δ , x 0 ) (\Delta ,x_0) is recomputed for the main theorem, which states that for all x ≥ x 0 x \geq x_0 there exists at least one prime in the interval ( x ( 1 − Δ − 1 ) , x ] (x(1-\Delta ^{-1}),x] .

A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

Abstract Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field K such that p is inert in K and that E has good reduction at p. Let $K_\infty$ be the anticyclotomic $\mathbb{Z}_p$-extension of K. Agboola–Howard defined Kobayashi-type signed Selmer groups of E over $K_\infty$ and showed that exactly one of them is cotorsion over the corresponding Iwasawa algebra. In this short note, we discuss a link between the characteristic ideals of the cotorsion signed Selmer group and the fine Selmer group building on a recent breakthrough of Burungale–Kobayashi–Ota on the structure of local points.

Generalizations of Bertrand’s Postulate to Sums of Any Number of Primes

Summary In 1845, Bertrand conjectured what became known as Bertrand’s postulate or the Bertrand-Chebyshev theorem: twice and prime strictly exceeds the next prime. Surprisingly, a stronger statement seems not to be well-known: the sum of any two consecutive primes strictly exceeds the next prime, except for the only equality . Our main theorem is a much more general result, perhaps not previously noticed, that compares sums of any number of primes. We prove this result using only the prime number theorem. We also give some numerical results and unanswered questions.

Characterizing the supernorm partition statistic

The paper gives characterizations of the supernorm statistic widehat{N} :mathcal {P}rightarrow mathbb {N}^{+} for partitions, where mathcal {P} is the set of integer partitions and mathbb {N}^{+} is the positive integers. The supernorm statistic map is a bijection onto mathbb {N}^{+} that defines a total order on partitions, the supernorm ordering, obtained by pulling back the additive total order on mathbb {N}^{+}. The supernorm ordering refines two partial orders on partitions: the multiset inclusion order, whose image under widehat{N} is the divisibility lattice on mathbb {N}^{+}, and the Young’s lattice order. The paper shows that it is characterized by these two properties, additionally with the requirement of an order-isomorphism from the multiset inclusion order to the divisibility order. It is also characterized by these two properties, additionally with the requirement of mapping the partitions with exactly one part bijectively to the prime numbers. It presents a construction showing that the latter additional conditions are necessary for the characterization to hold.

On the Divergence of the Sum of Prime Reciprocals

The sum of the reciprocals of all prime numbers diverges but the divergence is very slow. We propose effective lower and upper bounds for partial sums under the Riemann hypothesis. We give an explicit error term for all Mertens’ theorems.

On simply normal numbers with digit dependencies

AbstractGiven an integer and a set of prime numbers, the set of Toeplitz numbers comprises all elements of [0, b[ whose digits in the base‐b expansion satisfy for all and . Using a completely additive arithmetical function, we construct a number in that is simply Borel normal if, and only if, . We then provide an effective bound for the discrepancy.

Formula omitted]-graded identities of the Lie algebras U1

Let K be an infinite field of characteristic different from two and let U1 be the Lie algebra of the derivations of the algebra of Laurent polynomials K[t,t−1]. The algebra U1 admits a natural Z-grading. We provide a basis for the graded identities of U1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W1 of the derivations of the polynomial ring K[t]. The Z-graded identities for W1, in characteristic 0, were described in [8]. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in [8], and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.

A new combinatorial characterization of (quasi)-Hermitian surfaces

In this paper, we present a combinatorial characterization of a quasi-Hermitian surface as a set {mathcal {H}} of points of textrm{PG}(3,q), q=p^{2h}hge 1, p a prime number and qne 4, having the same size as the Hermitian surface and containing no plane, such that either a line is contained in {mathcal {H}} or intersects {mathcal {H}} in at most sqrt{q}+1 points and every plane intersects {mathcal {H}} in at least qsqrt{q}+1 points. Moreover, if there is no external line, the set {mathcal {H}} is a Hermitian surface.

A note on Friendly and Solitary Groups

In this paper, we extend the notions of friendly and solitary numbers to group theory and define friendly and solitary groups of type-1 and type-2. We provide many examples of friendly and solitary groups and study certain properties of the type-2 friends of cyclic $p$-groups, where $p$ is a prime number.

Blockchain Security Mechanism Design Based on Chinese Cryptosystem SM2 Algorithm

The cryptographic system is the foundation of blockchain security mechanisms. This paper constructs a blockchain security mechanism based on the Chinese national cryptographic SM2 algorithm, involving the generation of public–private key pairs and the signing of block information using key pairs. However, the original SM2 algorithm has a high computational complexity, which cannot meet the requirements of low on-chain delay in blockchain applications. To solve this problem, this paper optimizes the SM2 digital signature algorithm. The scalar multiplication operation in the digital signature algorithm has a large computational complexity, which affects the efficiency of a digital signature. To reduce its computational complexity, this paper improves the traditional window non-adjacent form (w-NAF) algorithm for the scalar multiplication operation and proposes a low computational complexity Fw-NAF algorithm based on the prime number precomputation window, which greatly reduces its computational complexity.

“Criticality” in the counting function of prime numbers: Theory and application to nuclear magic numbers

In this paper, considering critical phenomena and their phase transitions within the frame of the set of prime numbers, is attempted. Thus, the novel, theoretical and purely mathematical Model of Criticality based on Prime Numbers is introduced. This approach allows for the emergence of a parallelism between the physical concept of criticality and the corresponding concept in prime number theory. Based on this parallelism an application of the proposed model in determining the known magic numbers of Nuclear Physics is presented. This application introduces a physical meaning to the exceptional in properties set of the prime numbers and the corresponding prime number theory. Finally, going beyond the proposed model and its application, we suggest investigating other prime numbers as doubly magic numbers or candidate magic numbers for further experimental research.

KEMAMPUAN MATEMATIS MAHASISWA PGSD PADA MATA KULIAH RME MATERI JENIS BILANGAN

Mathematical abilities need to be possessed on the grounds of being able to equip life in solving problems regularly. The RME course bridges students to study mathematics in everyday life. The research was conducted to determine the mathematical thinking abilities of PGSD students in the RME course on number types. The type of research carried out was descriptive qualitative with the use of random sampling to determine 3 research subjects (high, medium and low mathematical abilities). The stages that were passed were the lecturer making questions, assessment criteria, conducting tests, scoring and assessment ending with data reduction assisted by MS Excel. The data obtained was that there were 16.67% students in the high mathematical ability category, 75% in the medium category and 8.33% in low mathematical ability. MS excel calculations show that the average value of students' mathematical ability is 52.78, so it is in the medium category. Students with high mathematical abilities are very good at understanding the material on odd numbers as well as irrational numbers, both on prime numbers and need guidance on prime numbers. Understanding of prime numbers is very good, good for even numbers as well as odd numbers but needs improvement in irrational numbers for students with moderate mathematical abilities. Students with low mathematical abilities have a good understanding of prime numbers, understand enough material for odd and even numbers and have no clue about irrational numbers.

Origin Point Must Represent Critical Line as Location for Nontrivial Zeros of Riemann Zeta Function, and Set Prime Gaps With Subsets Odd Primes Are Arbitrarily Large in Number

We prove the Countably Infinite Subsets of odd primes have cardinality of Arbitrarily Large in Number. This is achieved by demonstrating the asymptotic law of distribution of prime numbers that involves natural logarithm function to be applicable to all these subsets of odd primes derived from every even Prime gaps which are again Arbitrarily Large in Number. We prove the location for Countably Infinite Set of nontrivial zeros computed using Dirichlet eta function (proxy function for Riemann zeta function) is the unique critical line. This is achieved by plotting the infinitely many colinear lines of Riemann zeta function using both critical line and non-critical lines.

On certain inequalities for the prime counting function – Part III

As a continuation of [10] and [11], we offer some new inequalities for the prime counting function $\pi (x).$ Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given. Some results on $\pi (p_n^2)$ are offered, $p_n$ denoting the $n$-th prime number. Results on $\pi (\pi (x))$ are also considered.

Skew-polynomial-sparse matrix multiplication

Based on the observation that Q(p−1)×(p−1) is isomorphic to a quotient skew polynomial ring, we propose a new deterministic algorithm for (p−1)×(p−1) matrix multiplication over Q, where p is a prime number. The algorithm has complexity O(Tω−2p2), where T≤p−1 is a parameter determined by the skew-polynomial-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Here a matrix is skew-polynomial-sparse if its corresponding skew polynomial is sparse. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity O∼(tω−2p2+p2log⁡1ν), where t≤p−1 is the skew-polynomial-sparsity of the product and ν is the probability parameter. The main feature of the algorithms is the acceleration for matrix multiplication if the input matrices or their products are skew-polynomial-sparse.

Comprehensive Performance Study of Hashing Functions

Most literature on hashing functions speaks in terms of hashing functions being either ‘good’ or ‘bad’. In this paper, we demonstrate how a hashing function that gives good results for one key set, performs badly for another. We also demonstrate that, for a single key set, we can find hashing functions that hash the keys with varying performances ranging from perfect to worst distributions. We present a study on the effect of changing the prime number ‘$p$’ on the performance of a hashing function from $H_1$ Class of Universal Hashing Functions. This paper then explores a way to characterize hashing functions by studying their performance over all subsets of a chosen Universe. We compare the performance of some popular hashing functions based on the average search performance and the number of perfect and worst-case distributions over different key sets chosen from a Universe. The experimental results show that the division-remainder method provides the best distribution for most key sets of the Universe when compared to other hashing functions including functions from $H_1$ Class of Universal Hashing Functions.

The Chevalley–Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group)

Analytic theory of real abelian fields K says (in the cyclic semi-simple case) that the order of the p-class group HK is equal to the p-index of cyclotomic units (EK:FK). We develop, in this article, new promising links between: (i) the Chevalley–Herbrand formula giving the number of “ambiguous classes” in p-extensions L/K, L⊂K(μℓ) for auxiliary prime numbers ℓ≡1(mod2pN) inert in K; (ii) the phenomenon of capitulation of HK in L; (iii) the Main Theorem for the isotypic components using the irreducible p-adic characters φ of K, that we had conjectured (1977). We prove that this real Main Theorem is trivially fulfilled as soon as HK capitulates in L (Theorem 1.2). Computations with PARI programs support this new philosophy of the Main Theorem and the very frequent phenomenon of capitulation suggests Conjecture 1.1.

On primitivity and regularity of wreath product groups of degree 5p that are not p- groups using numerical approach

Let p be an odd prime number. This work applies some group concepts to construct the Wreath Product of two permutation groups of prime degrees. We used numerical approach to investigate and determine the primitive and regular nature of the constructed Wreath Product Group of degree 5p. We apply Computational Group Theory (GAP) to facilitate as well as validate our results.

New Discoveries in Number Theory and its Practical Application

This article shows new properties of positive integers related to the sum of the deviations of the products of dividing the positive number by each of its factors, as well as the positive integer with only three divisors called the perfect square with a square root as a prime number, and the article also shows the building scheme a computer application to find the number of divisors of a number, and to classify the number into prime and composite.

Solar Far Infrared-Induced Growth of Plants is Due to NonThermal Effect of Infrared to Radio-Frequency Electromagnetic Energy: Verification by Quantum Chemistry Molecular Modeling (DFT/MM)

This article shows new properties of positive integers related to the sum of the deviations of the products of dividing the positive number by each of its factors, as well as the positive integer with only three divisors called the perfect square with a square root as a prime number, and the article also shows the building scheme a computer application to find the number of divisors of a number, and to classify the number into prime and composite.