Sums Over Primes II

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

Some new semiring structures

In this paper, we introduce some new semiring structures by considering the prime factors, sum of divisors and the number of relatively prime positive divisors of a pair of non-negative integers and solve a variety of system of linear equations over one of these semirings. We then discuss the properties of these structures and how they compare with classical algebra. The paper is concluded by providing a key-exchange scheme using one of the newly discovered semirings.

On the Universal Encoding Optimality of Primes

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes follows from Goldbach’s conjecture, and is viewed as an upper efficiency limit for encoding any integer with the fewest possible additions. A consequence of the above is that primes optimally encode—multiplicatively and additively—all integers. Thus, the set P of primes is the unique, irreducible subset of ℤ—in cardinality and values—that optimally encodes all numbers in ℤ, in a factorial and summative sense. Based on these dual irreducibility/optimality properties of P, we conclude that primes are characterized by a universal “quantum type” encoding optimality that also extends to non-integers.

Binomial formulas via divisors of numbers

The purpose of this note is to prove several binomial-like formulas whose exponents are values of the function ω(n) counting distinct prime factors of n.

On the number of semismooth integers

Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].

On reduced rings with prime factors simple

We obtain a common generalization of the results by Wong and Birkenmeier-Kim-Park, respectively, which say that a reduced ring with unity is strongly (respectively, weakly) regular if and only if all of its prime homomorphic images are division rings (respectively, simple domains). Our arguments are different from those in the known proofs and are quite simple. They also give a characterization of weakly regular reduced rings without unity. This characterization implies in particular that the class of weakly regular reduced rings forms a radical class. However, even if a weakly regular reduced ring has no unity, its prime homomorphic images must be simple domains with unity. In the second part of the paper, we study reduced rings whose prime homomorphic images are simple domains (not necessarily with unity).

Safety of Food and Food Warehouse Using VIBHISHAN

Food is one of the integral parts of human life making the quality of food one of the prime factors in its selection for consumption. In order to maintain the food quality, it must be taken care of from the very first step where its quality may be affected, that is, warehouses. Food safety and safety of its warehouses is one of the major concerns, because many people lose their lives due to poor food quality. A robot that can ensure the safety of both food and warehouse can be one of the possible solutions, because taking care of huge warehouses is a tedious task and sometimes food present inside the warehouse gets unnoticed and thus get contaminated. Also safety of warehouses from intruders can be done by a robot, in any condition where it is difficult for human beings. This robot would be cheap and efficient and also make sure of safety, keeping the food intact and ensuring its fine quality.

Representasi verbal siswa sekolah menengah pertama dalam mengkomunikasikan faktor prima

The purpose of this study was to describe students' communication in solving prime factors using written and oral communication. This study uses a qualitative and descriptive approach. The subjects of this study were 46 seventh grade students of Simanjaya Lamongan, East Java, Indonesia, Middle School who had obtained prime factor material. Data analysis was obtained by using a test of prime factors and interviews. Based on the results of data analysis, it was concluded that: 1) Written communication, as many as 40 students were able to communicate in writing with a percentage of 87% and 6 students were unable to communicate in writing with a percentage of 13%; 2) Oral communication, at this stage interviews were conducted by taking the subject as many as two students taken from one student who is able to communicate in writing and one student who is not able to communicate in writing, the results obtained that students are able to communicate in writing are also able to communicate verbally. verbally, while students who are not able to communicate in writing, it turns out that students are able to communicate orally. From the explanation above, it can be concluded that students are able to explain concepts into mathematical language, students are able to explain mathematical arithmetic operations, students are able to explain mathematical solutions and students are able to convey ideas or opinions. Thus, students can communicate in writing and orally well in solving prime factors.