Perfect Numbers Research Articles

On the Within-Perfect Numbers

Abstract Motivated by the works of Erdös, Wirsing, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term ‘$(\ell;k)$-within-perfect numbers’, where $\ell \gt 1$ is a real number and $k: [1, \infty) \rightarrow (0, \infty)$ is an increasing and unbounded function.

A Perfect Infinite Series

Summary We show that the sum of the series ∑ n = 1 ∞ ( n 2 / 2 n ) is the perfect number 6, using simpler series with sums 1, 2, 3, which are the proper divisors of 6.

Optimization of Tax Collection Supervision in Increasing Local Original Revenue (PAD) by Gorontalo City Finance Agency

This study aims to determine the Optimization of Tax Collection Supervision in Increasing Regional Original Revenue (PAD) by the Gorontalo City Finance Agency. This research method uses descriptive qualitative research. The data collection techniques used are observation, interview, and documentation techniques. In accordance with the type of research used is qualitative analysis techniques. research results in the form of interviews conducted with research informants about Optimizing Supervision of Restaurant Tax Collection in Increasing Regional Original Revenue (PAD) by the Gorontalo City Finance Agency. The sub-focus used are: implementation standards, level of supervision, responsibility. From the statements or results of research with several informants above and from the observations of researchers, it shows that the standard implementation of restaurant tax collection by the Finance Agency has not been implemented optimally. Based on several statements or results of interviews with informants and from observations of researchers, it shows that the level of supervision carried out by the finance agency is still not optimal. From the results of interviews with several informants above, and from the observations of researchers, it shows that the responsibility of taxpayers is still a little less and has not reached a perfect number and the responsibility of the Finance Agency is also still not optimally carried out, this happens because there are still many taxpayers who do not understand the tools or sites they use in making transactions. This is one of the reasons why restaurant tax collection has not been optimized in increasing local revenue (PAD) in Gorontalo City.

Anticipating the long-term effects of daily weather conditions and socio-economic factors on FNN through metropolitan yard waste

The ability to accurately forecast the ability of a system to function is necessary for efficient and productive management of solid waste. The correctness of wastage forecast proposals, which have been the subject of extensive research, is tested. In contrast to wastage forecast proposals for varied waste products, factors for yard wastage are time-sensitive and the repercussions of delay must be considered. In this work, we focus on lag periods concerning factors that aim to forecast the creation of metropolitan yard wastage using the Deep Learning (DL) technique. Through correlation evaluation, socioeconomic and meteorological factors are evaluated on a daily average and the key factors are used to develop yard wastage proposals. Following that, these proposals employ Feed Forward Neural Networks (FNN), that variety of day measurements are represented by time-stamped variables. This facilitates to achieve the decrease of the daily yard wastage production's inaccuracy. We developed an FNN model that can forecast the influence of these factors on yard wastage generation and management methods using a wastage dataset that includes meteorological variables, socioeconomic indicators and distributions of yard wastage from a metropolis region. At the training and testing stages of yard wastage models, we compare the results using MSE, MAPE and R-Value at each stage's perfect number of layers. The effect of delay on the accurateness of daily yard wastage forecast proposals was highlighted by the MAPE of 25.35% that was attained during testing and the 23.2% reduction in the MSE displayed by yard wastage proposals during training.

Linear support for the prime number sequence and the first and second Hardy-Littlewood conjectures

Servais and Grün used results about linear support for the prime number sequence to obtain upper bounds on the smallest prime in odd perfect numbers. This was extended by Cohen and Hendy who proved that for every $n \in \mathbb {N}$ there exists an integer $b_n$ such that $p_i \ge p_1 + ni - b_n$ for any sequence $(p_i)$ of odd primes, and found minimal values of $b_n$ for $3 \le n \le 5$, and conjectured minimal values for $6 \le n \le 10$. We give a new proof of the existence of $b_n$ and the values for $3 \le n \le 5$. We also show that if we assume that the second Hardy-Littlewood conjecture, $\pi (a+b) \le \pi (a) + \pi (b)$ for $a, b \ge 2$, is true, then we can in a finite number of steps determine numbers, $T_n$, that give quite close bounds for the values of $b_n$, namely $T_n \le b_n \le T_n + n - 2$, and determine the values of $T_n$ for $6 \le n \le 20$. We also consider the question of whether the values of $b_n$ can be replaced by smaller numbers if we assume that $p_1 > 3$. We will show that if we assume that the first Hardy-Littlewood conjecture is true, then we can determine the minimum such values, $a_n$, for $3 \le n \le 5$. We also determine some lower bounds for $a_n$ for $n \ge 6$. It is well-known that the two Hardy-Littlewood conjectures are mutually exclusive, but we never use both conjectures simultaneously. These results give us the upper bound $p_1<\frac {n}{2^n-1}t+b_n$ on the smallest prime in odd perfect numbers, where $t$ is the number of distinct primes dividing the odd perfect number. This bound was also found by Cohen and Hendy, but we improve the constants $b_n$.

Application of Partitions of Odd Numbers and their Odd Sums to Prove the Nonexistence of Odd Perfect Numbers

Perfect numbers, which are integers equal to the sum of their proper divisors, excluding themselves, have intrigued mathematicians for centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p-1), where p and 2p-1 are prime numbers (Mersenne primes), the existence of odd perfect numbers remains an unsolved problem. This study aims to prove the nonexistence of odd perfect numbers by utilizing an algorithm which demonstrates that a positive even integer can be partitioned into all pairs of odd numbers. Using this approach, it is proven that any positive odd number 2n+1 can be partitioned into all pairs of both odd and even numbers and from the set of these partitions, we show that there exist a proper subset containing all proper divisors of 2n+1. Using these results, and the facts that there exist infinitely many odd numbers and the odd sums of odd numbers is always odd, we prove the nonexistence of odd perfect numbers contributing to the conjecture that they do not exist.

Each friend of 10 has at least 10 nonidentical prime factors

For each positive integer n, if the sum of the factors of n is divided by n, then the result is called the abundancy index of n. If the abundancy index of some positive integer m equals the abundancy index of n but m is not equal to n, then m and n are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.This work applies methods from Nielsen’s 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria.

All KnownPerfect Numbers other than 6 Satisfy N=4+6n

For all 51 known perfect numbers ranging from (p=2 to p= 82589933) and with the only exception of N=6, all perfect numbers belong to the group of natural numbers formed by N=4+6n. If this observation can be proven valid for all existing even perfect numbers, that would automatically exclude 2/3 of all even numbers out of the possibility of being perfect. If this can be proven a necessary condition for all perfect numbers, then it would rule out the possibility of having any odd perfect numbers.

Les nombres parfaits chez Mersenne : des mystères pythagoriques à la science et à l’exégèse

In responding to a question posed by Professor Vincent Carraud regarding a passage in a letter by Etienne Pascal in which he mentions an author who speaks of the mysteries of Pythagorean numbers we have resumed our dossiers on number theory in Mersenne. We have pointed to texts from Mersenne’s correspondence, printed books, and unpublished theological manuscripts between 1638 and 1648 in which the mysteries of the Pythagorean numbers are mentioned with reference to the theory of perfect numbers and with reference to Peter Bongo’s work, which was widespread throughout Europe and particularly in France, De numerorum mysteria (1599 ed. aucta). This has allowed us to provide some indications taken from Mersenne’s unpublished manuscripts in which he addresses the mathematical problem of perfect numbers in exegesis, according to the new dating of his unpublished theological manuscripts that we have established.

A note on the Diophantine equations [formula omitted] and application to odd perfect numbers

Let N be an odd perfect number. Then, Euler proved that there exist some integers n,α and a prime q such that N=n2qα, q∤n, and q≡α≡1mod4. In this note, we prove that the ratio σ(n2)qα is neither a square nor a square times a single prime unless α=1. It is a direct consequence of a certain property of the Diophantine equation 2ln2=1+q+⋯+qα, where l denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders Z[1−q] and the primitive solutions of generalized Fermat equations xβ+yβ=2z2. We give also a slight generalization to odd multiply perfect numbers.

On a theorem of Kanold on odd perfect numbers

On a theorem of Kanold on odd perfect numbers

On the index of special perfect polynomials

We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $\omega(d) \geq 9$, and $\deg(d) \geq 258$, where $d := \gcd(Q^2,\sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ \sigma(A)=A$ in the polynomial ring ${\mathbb{F}}_2[x]$. The function $\sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].

On the Picture-Perfect Number

This is a seminar paper on the article picture perfect numbers by Joseph L. Pe that was published in the journal mathematical spectrum in 2008. This paper begins with a discussion of the definition of the more familiar concept of perfect numbers, then proceeds to a discussion of the picture-perfect numbers as defined by Joseph L. Pe, and winds up with a discussion on how to obtain a picture-perfect number using the Andersen's Theorem. This paper also includes proof of the Andersen's Theorem, as well as that of the Andersen's Lemma, which are both attributed to Jens Kruce Andersen.

An Upper Bound for Odd Perfect Numbers

An Upper Bound for Odd Perfect Numbers

Raphael‟s “The School of Athens”: A Hypothesis on the Identity and Symbolic Meaning of the Figure in White Cloak: The True Muse of Raphael

Raphael’s fresco “The School of Athens” is rich with figures of prominent philosophers who are identified by their certain characteristic details. The identity of a figure in Raphael’s “The School of Athens” Plato group traditionally attributed to Francesco Rovere, is currently being questioned. Present article brings forward a new hypothesis on this matter: the aforementioned figure is seen as a collective figure that symbolizes the unity of a man and a woman, in this case, Raphael and his muse. Besides, this figure also encodes the year when the fresco was painted. As a key to the code, the blackboard containing the Platonic perfect number 10 immediately at the figure’s feet is important, as well as the positions of some of the figures in the first row on the left part of the painting. The article also makes some revelations about Raphael’s real muse, who, according to the author, was the courtesan Imperia. Evidence of this is the fact that Imperia is portrayed in many of Raphael’s works.

NEW ARITHMETIC FUNCTION RELATED TO THE LEAST COMMON MULTIPLE

The author’s last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers.

Corrigendum to: “Some modular considerations regarding odd perfect numbers – Part II” [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24

In [2], the authors proposed a theorem which they recently found out to contradict Chen and Luo’s results [1]. In the present paper, we provide the correct form of this theorem.

NOTE ON THE TWO CELEBRITY NUMBERS, PERFECT NUMBERS AND TRIANGULAR NUMBERS

Mathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. Thus a positive integer is called a perfect number if it is equal to the sum of its proper positive divisors. The search for perfect numbers began in ancient times. The four perfect numbers 6, 28, 496, and 8128 seem to have been known from ancient times. In this paper, we will investigate some important properties of perfect numbers. We give easy and simple proofs of theorems using finite series. We give our own alternative proof of the well-known Euclid’s Theorem
 (Theorem I). We will also prove some important theorems which play key roles in the mathematical theory of perfect numbers.

The Costliness of US Residency Applications: Moving Toward Preference Signaling and Caps.

The Costliness of US Residency Applications: Moving Toward Preference Signaling and Caps.

On the distribution of perfect numbers and related sequences via the notion of the disc

On the distribution of perfect numbers and related sequences via the notion of the disc