There not exists odd perfect numbers

2022

We prove that every odd perfect number N has at least 420 distinct prime divisors, and that N is greater than 1~ 9 x 10'""'. 0 AcadCmie des Sciences/Elsevier, Paris I1 n'existe pas de petits nombres par$aits impairs RCsumC. Nous defmontrons que tout nombre N parfait impair posskde au mains 420 facteurs premiers d@rents, et que N est plus grand que 1,9 x 10""50. 0 AcadCmie des SciencesElsevier, Paris Version francaise abrt!gtSe Un entier positif n est appelC pa&it si c(n) = 2n, oti (T dksigne la fonction somme des diviseurs, i.e. cr(n) = Cd, n d. On ignore s'il existe des nombres parfaits impairs. Des remarques gCnCrales ainsi que des rCf&ences concernant ce probkme peuvent &tre trouvkes en [9], [lo]. Cette Note concerne le cas oti il existe des nombres parfaits impairs, et N dksigne un tel nombre.

In this article we solve one of the oldest and celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there be no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all even perfect numbers. We answer the question of whether there exists an odd perfect number in the negative by proving a theorem asserting that the existence of such a number would lead to contradictions (proof by reductio ad absurdum). Somewhat remarkably, perhaps, this result is proved using only elementary methods. Hence, the popular conjecture that odd perfect numbers do not exist, no matter how large these numbers might be, is confirmed to be correct. Thus, one of the oldest and celebrated questions in mathematics has now a definitive answer.

In this article we solve one of the oldest and celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there be no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all even perfect numbers. We answer the question of whether there exists an odd perfect number in the negative by proving a theorem asserting that the existence of such a number would lead to contradictions (proof by reductio ad absurdum). Somewhat remarkably, perhaps, this result is proved using only elementary methods. Hence, the popular conjecture that odd perfect numbers do not exist, no matter how large these numbers might be, is confirmed to be correct. Thus, one of the oldest and celebrated questions in mathematics has now a definitive answer.

ON ODD PERFECT NUMBERS, 2021

In this note, we introduce the notion of the disc induced by an arithmetic function and apply this notion to the odd perfect number problem. We show that no odd perfect numbers exist by exploiting this concept.

Theorem. The square of an odd prime number cannot be very perfect number.

Notes on Number Theory and Discrete Mathematics, 2018

In this note, we revisit and show how some old results on odd perfect numbers follow from assuming some unproven yet reasonable conjectures.

A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as "Odd Near-Perfect Numbers" and "Deficient-Perfect Numbers". Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 10 1500 , The total number of prime factors/divisors of an odd perfect number is at least 101, and 10 8 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.

Academia Biology, 2023

NThe polar regions are home to a great biodiversity of vertebrates that are highly impacted by climate change. As extreme and pristine ecosystems, the use of environmental DNA (eDNA) presents itself as a noninvasive tool for the monitoring of polar vertebrates’ species, essential for their preservation. In this article, we review the currently published research on vertebrate eDNA in the Arctic, as well as compare it with a similar already-published article on Antarctica. Our results show how the Artic region is well represented with several studies focusing on vertebrates, being the second most researched group of organisms (19%), only behind plants, and with several studies related to the development of new approaches and methodologies for the use of eDNA from cold regions being published in the last years. In opposite, the Antarctic eDNA science on vertebrates is still poorly developed, with the vast majority of studies focusing on microbial life; therefore, the Antarctic region still remains with a great potential for the research development in vertebrates and the effects of the past and recent climate changes on their populations