# EXTENSIONS OF SOME RESULTS CONCERNING ODD PERFECT NUMBERS

@inproceedings{Williams1985EXTENSIONSOS, title={EXTENSIONS OF SOME RESULTS CONCERNING ODD PERFECT NUMBERS}, author={R. J. Williams}, year={1985} }

for distinct odd primes p, ql9 ..., qt, with p = a = 1 (mod 4). (We shall always assume this form for the prime factor decomposition of N) . Many writers have found conditions which must be satisfied by the exponents 23i» ...5 2(3t» and it is our intention here to extend some of those results. We shall find it necessary to call on a number of conditions of other types, some of which have only recently been found. These are outlined in Section 2. It is known (see [8]) that we cannot have ^ E 1…

## 15 Citations

### NONEXISTENCE OF ODD PERFECT NUMBERS OF A CERTAIN FORM NONEXISTENCE OF ODD PERFECT NUMBERS OF A CERTAIN FORM

- Mathematics
- 2009

Write N = pαq1 1 · · · q 2βk k , where p, q1, . . . , qk are distinct odd primes and p ≡ α ≡ 1 (mod 4). An odd perfect number, if it exists, must have this form. McDaniel proved in 1970 that N is not…

### On the total number of prime factors of an odd perfect number

- MathematicsMath. Comput.
- 2003

It is proved that if βj ≡ 1 (mod 3) orβj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3|n is perfect, where σ(n) denotes the sum of the positive divisors of n.

### Necessary Conditions For the Non-existence of Odd Perfect Numbers

- Mathematics
- 2005

We start with a result showing most odd cubes cannot be perfect numbers (see Theorem 1). Then we give a new proof of a special case of a result of Iannucci (see [IAN]) that shows that none of the…

### Quasiperfect Numbers With the Same Exponent

- MathematicsIntegers
- 2019

A general lower bound for the smallerst prime factor of quasiperfect numbers is found and some upper bounds concerning quas Piperfect numbers of the form $N=m^2$ with $m$ squarefree are found.

### On the divisibility of odd perfect numbers by a high power of a prime

- Mathematics
- 2005

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite…

### Solving the Odd Perfect Number Problem: Some Old and New Approaches

- Mathematics
- 2012

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs)…

### On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime

- Mathematics
- 2016

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1,…

### An exponential diophantine equation related to odd perfect numbers

- Mathematics
- 2018

We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give…

### ON A VARIATION OF PERFECT NUMBERS

- Mathematics
- 2006

We define a positive integer n to be k-imperfect if kρ(n) = kn for some integer k ≥ 2. Here, ρ is a multiplicative arithmetic function defined by ρ(pa) = pa − pa−1 + pa−2 − · · · + (−1)a for a prime…

### Odd multiperfect numbers

- Mathematics
- 2011

A natural number $n$ is called {\it multiperfect} or
{\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where
$\sigma(n)$ is the sum of the positive divisors of $n$. In this
paper, we…

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