scholarly journals On Odd Deficient-Perfect Numbers with Four Distinct Prime Divisors

2016 ◽  
Vol 06 (05) ◽  
pp. 411-417
Author(s):  
兰 崔
Keyword(s):  
Prime Divisors ◽  
Perfect Numbers

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 A rationality condition for the existence of odd perfect numbers

2003 ◽  
Vol 2003 (20) ◽  
pp. 1261-1293 ◽  
Author(s):  
Simon Davis
Keyword(s):  
Lower Limit ◽  
Upper Bound ◽  
Fixed Interval ◽  
Square Root ◽  
Prime Divisors ◽  
Rationality Condition ◽  
Nonexistence Of Solutions ◽  
Perfect Numbers ◽  
Special Case

A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such thatσ(N)could be equal to2N, whereNbelongs to a fixed interval with a lower limit greater than10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and it is shown that the arithmetic primitive factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation. This equation is one example of a more general prime equation,(qjn−1)/(qin−1)=ph, and the demonstration of the nonexistence of solutions whenh≥2requires the proof of a special case of Catalan's conjecture. General theorems on the nonexistence of prime divisors satisfying the rationality condition and odd perfect numbersNsubject to a condition on the repunits in factorization ofσ(N)are proven.

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.

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

Perfect Numbers

1910 ◽  
Vol 17 (8-9) ◽  
pp. 165-168
Author(s):  
T. M. Putnam
Keyword(s):  
Perfect Numbers

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

65.1 Even Perfect Numbers

1981 ◽  
Vol 65 (431) ◽  
pp. 28 ◽  
Author(s):  
Graeme L. Cohen
Keyword(s):  
Perfect Numbers
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