Are There Numbers That Are Slightly Excessive?

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Homework Help Overview

The discussion revolves around the concept of slightly excessive numbers, also referred to as quasiperfect numbers, and their properties in relation to their divisors. Participants explore whether such numbers exist, particularly focusing on their factor pairs and the implications of their definitions.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants examine the definitions of quasiperfect numbers and question the implications of factors being odd or even. There are attempts to clarify the relationship between the sum of divisors and the nature of these numbers, with some participants expressing confusion over the definitions and properties involved.

Discussion Status

The discussion is ongoing, with various interpretations being explored regarding the nature of quasiperfect numbers. Some participants have offered insights into the definitions and properties, while others are questioning assumptions and seeking clarification on specific points.

Contextual Notes

There is a noted confusion regarding the distinction between proper divisors and all divisors, which affects the understanding of the properties of quasiperfect numbers. Additionally, references to existing mathematical literature and conjectures about the nature of these numbers are present, indicating a complex background to the discussion.

  • #31
It should be \sigma(2^k)=2^{k+1}-1. this tells youe 2^k can't be quasiperfect right?

So what about 2^k n where n is odd? If this is quasiperfect, can you say anything about n?
 
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  • #32
genii not geniuses said:
if k = 4 then sigma(2^k) = 15

16 is a divisor of 16.
 
  • #33
I follow you now but I don't know what I can say about n.
 
  • #34
Can you say anything about \sigma(2^k n)? You know it must be odd if this is quasiperfect...
 
  • #35
well sigma(2^k*n) is even.
 
  • #36
genii not geniuses said:
well sigma(2^k*n) is even.

What makes you say that?
 
  • #37
Well, isn't the sum of the divisors of an even number always even?
 
  • #38
genii not geniuses said:
Well, isn't the sum of the divisors of an even number always even?

No...\sigma(2)=3, \sigma(18)=\sigma(2)\sigma(9)=3\times 13=39
 
  • #39
Ohhhhhh... I'm used to not counting the number itself, which is why I keep getting confused.
 
  • #40
sigma(2^k * n) = odd
 
  • #41
genii not geniuses said:
sigma(2^k * n) = odd

\sigma(6)=12

get used to counting the number itself. This sigma I've defined is the standard version across number theory texts.

You know that \sigma(ab)=\sigma(a)\sigma(b) when a and b are relatively prime right (i.e. sigma is multiplicative)? Use this fact here.

I'm thinking we may be in trouble later on though, do you know about Quadratic Reciprocity?
 
  • #42
No, explain please!
 
  • #43
genii not geniuses said:
No, explain please!

I think you should get yourself a book with "elementary/introductory number theory" in the title and get to work. Head over to the "Number Theory" section of this website, there's a few threads posting book suggestions. get one (or several) and post questions there. This isn't really the best format for several number theory lectures (nor do i have the time), but we can definitely help you with sticking points as they come up.
 
  • #44
ok, thanks!
 

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