Perfect Numbers Research Articles

Some modular considerations regarding odd perfect numbers

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $\sigma(m^2)/p^k$ is a square.

Acting in the public interest: accounting for the vulnerable

This article seeks to initiate research around the potential roles of the accounting profession for tackling the challenges of the vulnerable. Its backdrop is the current consideration of the profession’s public interest role. The importance of dialogue around the public interest role is evidenced by the increasing levels of vulnerability, even within developed countries. Accounting underpinned by broader values has potential to provide knowledge of issues relating to the vulnerable. However, the accounting profession has only engaged with such potential to a limited degree. The article overviews existing knowledge and areas within which more research is required. In order to illustrate the potential for such research, initial findings from two case studies of homelessness (an example of the vulnerable) provide evidence as to the importance, and challenges, of accounting for the vulnerable. This article highlights the need to: take a principles-based approach in defining the vulnerable, undertake an accounting that reflects the lives they value, acknowledge that there are different ways for addressing these issues, recognise that an absence of perfect numbers should not become a barrier to action, and that accounting for the vulnerable is one way that the accounting profession may discharge their public interest roles.

Some properties of Zumkeller numbers and k-layered numbers

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer n is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be σ(n)/2. Generalizing even further, we call n a k-layered number if its divisors can be partitioned into k sets with equal sum.In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize k-layered numbers with two distinct prime factors and even k-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed.

On a Relation between the Perfect Roman Domination and Perfect Domination Numbers of a Tree

A dominating set in a graph G is a set of vertices S ⊆ V ( G ) such that any vertex of V − S is adjacent to at least one vertex of S . A dominating set S of G is said to be a perfect dominating set if each vertex in V − S is adjacent to exactly one vertex in S. The minimum cardinality of a perfect dominating set is the perfect domination number γ p ( G ) . A function f : V ( G ) → { 0 , 1 , 2 } is a perfect Roman dominating function (PRDF) on G if every vertex u ∈ V for which f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γ R p ( G ) . In this paper, we prove that for any nontrivial tree T, γ R p ( T ) ≥ γ p ( T ) + 1 and we characterize all trees attaining this bound.

Some New Notes on Mersenne Primes and Perfect Numbers

<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

Компьютер как новая реальность математики: III. Числа Мерсенна и суммы делителей

Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 2500 years ago. A description of the trajectories of this function — perfect numbers, amicable numbers, sociable numbers, and the like — constitute the contents of several problems stated over 2500 years ago, which still seem completely inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily, than what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson

An arithmetical question related to perfect numbers

For n ≥ 1 an integer, we shall denote by σ(n) the sum of the (positive) divisors of n.

On the Euler’s Form of an odd Perfect Number

Euler has proved that an odd perfect number, if exists, must be of the form ... , ≡ ≡ 1 (mod 4). In this article, we show: (i) An alternative proof to the Euler’s form of odd perfect numbers. (ii) An odd number of the form: , ≡ ≡ 1 (mod 4) cannot be perfect.

Corrigendum to the paper “A new upper bound for odd perfect numbers of a special form” (Colloq. Math. 156 (2019), 15–23)

We fix an error in the paper of the title. An erroneous citation in the references is also corrected.

On bounds of perfect number

On bounds of perfect number

On near-perfect numbers

The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and

Perfect Pell and Pell–Lucas numbers

Abstract The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

Developing and Implementing a Digital Navigation Program to Improve Outcomes for Medicare Bundle Patients Undergoing Joint Replacement Surgery

Background Evidence-based patient education and consistent, timely communication is key to ensuring good outcomes among joint replacement patients. Mount Sinai Hospital (MSH) participates in the mandatory CMS bundle for comprehensive joint replacement (CJR). MSH’s bundled payment strategy focuses on the development of a standardized model of care, built around evidence-based best practices to achieve the triple aims of strengthening population health while controlling cost and improving the quality of care. MSH launched a comprehensive digital navigation program (DNP) to guide joint replacement patients and their caregivers through pre-surgical preparation and recovery. Objective The objective was to improve the quality of care for joint replacement patients through creation of a digital navigation program specifically tailored to Medicare patients (age 65+) across the continuum of care. Methods Mount Sinai App Lab, in collaboration with the Department of Orthopedics, developed three digital therapeutic modules that were delivered through the RxUniverse Digital Medicine platform (Rx.Health, NY). These automated messages, programmed to send at specific times, included exercise instructions, medication reminders, and suggestions for how to prepare the home for optimal recovery. Messages specifically targeted key patient outcomes: length of stay, readmissions, ambulation on postoperative day 0, and discharge disposition. Staff “prescribed” each module to patients to ensure that engagement aligned with their process of care. Results Clinicians, patients, and their caregivers were receptive to the DNP. Patients showed a high rate of engagement, clicking links to educational content 873 times, and patients called their case manager or surgeon’s office to clarify their next steps when prompted. After 9 months, clinical outcomes for the digital navigation program were compared to other Medicare patients who had not received it. DNP patients had significantly shorter length of stay than their peers (2.81 vs 4.31 days). They also had a lower readmission rate (1.9% vs 2.9%) as well as a higher rate of discharge to home (87.8% vs 64.3%) and were more likely to ambulate on the day of surgery (47.9% vs 33.3%). Twenty patients responded to an end-of-program survey about their experience; 18 patients (90%) agreed that the program was helpful with the process of their total joint replacement surgery, 2 patients neither agreed nor disagreed (10%), and 0 patients disagreed. When asked about their satisfaction with the message volume, 19 patients answered “yes, this was the perfect number,” (95%) and 1 patient answered “no, I want fewer messages” (5%). Patient qualitative feedback was very positive. Patients reported, “Texts reassured me and helped me along with my progress and recovery,” and “Good support. Thank you.” A third said, “The texting program serves as a great reminder as what to do and when.” Conclusions The CJR DNP provided a direct, automated channel to educate and support patients at each stage of care. It demonstrated that digital navigation technology can be used even among non-digital native populations. It resulted in a significantly reduced length of stay and hospital readmissions among participating patients. Next steps include scaling the program across the health system and adding Spanish-language support. Similar programs are now being implemented for other surgical and disease use cases.

An analog of perfect numbers involving the unitary totient function

We shall give some results for an integer divisible by its unitary totient.

Upper bounds on the second largest prime factor of an odd perfect number

Acquaah and Konyagin showed that if [Formula: see text] is an odd perfect number with prime factorization [Formula: see text], where [Formula: see text], then one must have [Formula: see text]. Using methods similar to theirs, we show that [Formula: see text] and that [Formula: see text] We also show that if [Formula: see text] and [Formula: see text] are close to each other, then these bounds can be further strengthened.

Deficient perfect numbers with four distinct prime factors

For a positive integer [Formula: see text], if [Formula: see text] denotes the sum of the positive divisors of [Formula: see text], then [Formula: see text] is called a deficient perfect number if [Formula: see text] for some positive divisor [Formula: see text] of [Formula: see text]. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

The Perfect Numbers of Pell Number

For any positive integer n, let Pn, Qn denote the Pell sequence and Pell Lucas sequence respectively. The main purpose of this paper is using the elementary methods to study that Pn does not contain any even perfect numbers, Qn just contains one perfect number and this number is Q2 = 6.

Perfect and Deficient Perfect Numbers

For n a positive integer, let denote the sum of the divisors of n. The number n is said to be deficient perfect if for some . We show that every perfect number has a deficient perfect divisor, generalizing a consequence of Euclid and Euler’s characterization of even perfect numbers, namely that every even perfect number has an almost perfect divisor.

Re-examination of the different origins of the arithmetical books of Euclid's Elements

A quick examination of the diagrams in the Greek manuscripts of Euclid's Elements shows that VII.28 is the only proposition which has a diagram with both horizontal and vertical lines, and it is quite similar to parallel propositions in Book X (X.15, 16). This leads to an audacious assumption that all the propositions of Book VII after it may have been added later, and their authenticity is examined by logical and linguistic analysis.In many of the propositions of Book VII after VII.28, we have found characteristics or idiosyncrasies which can be explained by the hypothesis of later addition. Moreover, some propositions can be interpreted as lemmata for conspicuous results in Book IX such as IX.20 (the number of prime numbers surpasses any given number), IX.36 (sufficient condition of a perfect number).While the prevalent view is that Book VII is a well-organized basic theory from which various arguments in the subsequent two books are developed, our examination suggests that Book VII itself contains heterogeneous components which are intended for specific arguments in subsequent books. Further investigation is needed to find out whether this heterogeneity in Book VII is the result of later intervention, or that of original compilation.

On perfect Roman domination number in trees: complexity and bounds

A perfect Roman dominating function on a graph $$G =(V,E)$$ is a function $$f: V \longrightarrow \{0, 1, 2\}$$ satisfying the condition that every vertex u with $$f(u) = 0$$ is adjacent to exactly one vertex v for which $$f(v)=2$$ . The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted by $$\gamma _{R}^{p}(G)$$ , is the minimum weight of a perfect Roman dominating function in G. In this paper, we first show that the decision problem associated with $$\gamma _{R}^{p}(G)$$ is NP-complete for bipartite graphs. Then, we prove that for every tree T of order $$n\ge 3$$ , with $$\ell $$ leaves and s support vertices, $$\gamma _R^P(T)\le (4n-l+2s-2)/5$$ , improving a previous bound given in Henning et al. (Discrete Appl Math 236:235–245, 2018).