scholarly journals Some New Notes on Mersenne Primes and Perfect Numbers

2020 ◽  
Vol 3 (1) ◽  
pp. 15
Author(s):  
Leomarich F Casinillo
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Mathematical Structure ◽  
Prime Numbers ◽  
Perfect Number ◽  
Positive Divisor ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

<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&amp;space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;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&amp;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&amp;space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;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&amp;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>

Pseudoperfect numbers with no small prime divisors

2009 ◽  
Vol 93 (528) ◽  
pp. 404-409
Author(s):  
Peter Shiu
Keyword(s):  
Difficult Problem ◽  
Prime Divisors ◽  
Perfect Number ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

scholarly journals Deficient perfect numbers with four distinct prime factors

2019 ◽  
Vol 13 (07) ◽  
pp. 2050126 ◽  
Author(s):  
Parama Dutta ◽  
Manjil P. Saikia
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Positive Divisor ◽  
Prime Factors ◽  
Perfect Numbers

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 Fifth Unitary Perfect Number

1975 ◽  
Vol 18 (1) ◽  
pp. 115-122 ◽  
Author(s):  
Charles R. Wall
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Unitary Divisor ◽  
Perfect Numbers ◽  
Image Position

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] thatis also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.

Investigating Prime Numbers and the Great Internet Mersenne Prime Search

2002 ◽  
Vol 8 (2) ◽  
pp. 70-76
Author(s):  
Jeffrey J. Wanko ◽  
Christine Hartley Venable
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Prime Number ◽  
Prime Numbers ◽  
School Students ◽  
Large Numbers ◽  
Mersenne Prime

Middle school students learn about patterns, formulas, and large numbers motivated by a search for the largest prime number. Activities included.

scholarly journals Polygonal Numbers

2013 ◽  
Vol 21 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Related Result ◽  
Integer Sequences ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Polygonal Numbers ◽  
Online Encyclopedia ◽  
Four Square ◽  
Mersenne Primes ◽  
Ordinal Index

Summary In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

scholarly journals Thirty-nine perfect numbers and their divisors

1986 ◽  
Vol 9 (1) ◽  
pp. 205-206 ◽  
Author(s):  
Syed Asadulla
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Triangular Number ◽  
Perfect Numbers

The following results concerning even perfect numbers and their divisors are proved: (1) A positive integernof the form2p−1(2p−1), where2p−1is prime, is a perfect number; (2) every even perfect number is a triangular number; (3)τ(n)=2p, whereτ(n)is the number of positive divisors ofn; (4) the product of the positive divisors ofnisnp; and (5) the sum of the reciprocals of the positive divisors ofnis2. Values ofpfor which 30 even perfect numbers have been found so far are also given.

A Relation Between Triangular Numbers and Prime Numbers

Integers ◽  
2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Rigoberto Flórez ◽  
Leandro Junes
Keyword(s):  
Positive Integer ◽  
Prime Numbers ◽  
Triangular Numbers

Abstract.We study a relation between factorials and their additive analog, the triangular numbers. We show that there is a positive integer

scholarly journals Perfect numbers and Fibonacci primes (I)

2014 ◽  
Vol 11 (01) ◽  
pp. 159-169
Author(s):  
Tianxin Cai ◽  
Deyi Chen ◽  
Yong Zhang
Keyword(s):  
Positive Integer ◽  
Perfect Number ◽  
Perfect Numbers

In this paper, we introduce the concept of F-perfect number, which is a positive integer n such that ∑d|n,d<n d2 = 3n. We prove that all the F-perfect numbers are of the form n = F2k-1 F2k+1, where both F2k-1 and F2k+1 are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.

3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

ON DEFICIENT-PERFECT NUMBERS

2014 ◽  
Vol 90 (2) ◽  
pp. 186-194 ◽  
Author(s):  
MIN TANG ◽  
MIN FENG
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Perfect Number ◽  
Perfect Numbers

AbstractFor a positive integer $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma (n) = 2n - d$. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.

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