Prime Cubes Research Articles

A classification of regular maps with Euler characteristic a negative prime cube

A classification of regular maps with Euler characteristic a negative prime cube

A New Heuristic Tool for Designing Multiple-Valued Logic Systems

In this paper, a brand new heuristic optimization algorithm for multiple-valued logic (MVL) functions and the performance of the implementation will be introduced. In logic synthesis, optimization procedure is a fundamental process in order to design more efficient systems. The proposed algorithm, called MVL-MIN, utilizes Cube Algebra operators. MVL-MIN computes the prime cubes that cover a target cube at a time, instead of extracting all the primes for a given MVL function. MVL-MIN consists of five major procedures, called (a) checking, (b) expansion, (c) elimination, (d) nondisjoint sharp, and (e) prime determination. In order to find a near-minimal cover, three different prime determination functions are developed. The implementation based on the new algorithm is compared with MVSIS. For testing, three sets of test bench are prepared. Each set includes 5 MVL functions. Test results proved that MVL-MIN is able to solve all test files within a fixed allocation of computer resources while MVSIS could not. MVL-MIN achieved 1.9× to 9.7× speedup over MVSIS. In terms of cover size, since current MVL-MIN implementation cannot recognize redundant cubes, MVSIS computed more concise covers for all test benches than MVL-MIN.

Powers of 2 in two Goldbach-Linnik type problems involving prime cubes

In this paper, we show that every sufficiently large even integer can be represented as eight prime cubes and k1 powers of 2, and every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k2 powers of 2, where k1=28 and k2=98. These results improve previous results k1=30 and k2=335, respectively.

Sums of One Prime Power and Four Prime Cubes in Short Intervals

Let k ⩾ 1 be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers n = p 1 k + p 2 3 + p 3 3 + p 4 3 + p 5 3 in short intervals, where p 1 , p 2 , p 3 , p 4 , p 5 are prime numbers.

A pair of equations in two prime squares, four prime cubes and powers of two

A pair of equations in two prime squares, four prime cubes and powers of two

On pairs of equations in eight prime cubes and powers of 2

<abstract><p>In this paper, it is proved that every pair of large positive even integers satisfying some necessary conditions can be represented in the form of a pair of eight cubes of primes and 287 powers of $ 2 $. This improves the previous result.</p></abstract>

A PAIR OF EQUATIONS IN EIGHT PRIME CUBES AND POWERS OF 2

AbstractIn this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$ . In particular, we prove that $k=335$ is admissible, which improves the previous result.

Two prime squares, four prime cubes and powers of 2

Two prime squares, four prime cubes and powers of 2

Two results on Goldbach–Linnik problems for cubes of primes

It is proved that every pair of sufficiently large even integers can be represented in the form of a pair of equations of eight prime cubes and 609 powers of 2, and each sufficiently large even integer is the sum of eight cubes of primes and 157 powers of 2. These results constitute refinements upon those of Z. X. Liu (2013) and of X. D. Zhao and W. X. Ge (2020).

Exceptional Sets for Sums of Prime Cubes in Short Interval

Let E N , X denote the number of even integers n , with N − X ≤ n ≤ N , such that n cannot be written as n = p 1 3 + ⋯ + p 8 3 . We prove that if X > N 1 / 36 + ɛ , then E N , X = o X .

Goldbach–Linnik type problems on cubes of primes

In this paper, we will investigate three Goldbach–Linnik type problems on cubes of primes. For example, it was proved that, for $$k=1364,$$ every pair of sufficiently large even integers can be represented by a pair of eight prime cubes and k powers of 2. In this paper, we give the detailed proof for the first time and sharpen the value of k to 658.

A pair of equations in unlike powers of primes and powers of 2

Abstract In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.

ON SUMS OF TWO PRIME SQUARES, FOUR PRIME CUBES AND POWERS OF TWO

We prove that every sufficiently large even integer can be represented as the sum of two squares of primes, four cubes of primes and 28 powers of two. This improves the result obtained by Liu and Lü [‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131(4) (2011), 716–736].

The density of integers representable as the sum of four prime cubes

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period p n + 1 , where p = d R + 1 is an odd prime, d , R are two non-negative integers, and n > 0 is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period p n + 1 ( n ≥ 0 ) with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

Sums of four prime cubes in short intervals

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.

Exceptional sets for sums of almost equal prime cubes

Abstract In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

A novel switching function approach for data mining classification problems

Rule induction (RI) is one of the known classification approaches in data mining. RI extracts hidden patterns from instances in terms of rules. This paper proposes a logic-based rule induction (LBRI) classifier based on a switching function approach. LBRI generates binary rules by using a novel minimization function, which depends on simple and powerful bitwise operations. Initially, LBRI generates instance codes by encoding the dataset with standard binary code and then generates prime cubes (PC) for all classes from the instance codes by the proposed reduced offset method. Finally, LBRI selects the most effective PC of the current classes and adds them into the binary rule set that belongs to the current class. Each binary rule represents an If–Then rule for the rule induction classifiers. The proposed LBRI classifier is based on basic logic functions. It is a simple and effective method, and it can be used by intelligent systems to solve real-life classification/prediction problems in areas such as health care, online/financial banking, image/voice recognition, and bioinformatics. The performance of the proposed algorithm is compared to six rule induction algorithms; decision table, Ripper, C4.5, REPTree, OneR, and ICRM by using nineteen different datasets. The experimental results show that the proposed algorithm yields better classification accuracy than the other rule induction algorithms on ten out of nineteen datasets.

Two prime squares, four prime cubes and powers of 2

Two prime squares, four prime cubes and powers of 2

Tetravalent half‐arc‐transitive bi‐p‐metacirculants

AbstractA graph is half‐arc‐transitive if its full automorphism group acts transitively on its vertex set and edge set, but not arc set. A graph is said to be a bi‐Cayley graph over a group if it admits as a group of automorphisms acting semiregularly and with two orbits on the vertex set. In this paper, a classification is given of tetravalent half‐arc‐transitive bi‐Cayley graphs over metacyclic ‐groups for each odd prime , and this is then used to give a classification of tetravalent half‐arc‐transitive graphs of order twice a prime cube.