Are There Numbers That Are Slightly Excessive?

[shmoe]

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?

 

[shmoe]

if k = 4 then sigma(2^k) = 15

16 is a divisor of 16.

 

[genii not geniuses]

I follow you now but I don't know what I can say about n.

 

[shmoe]

Can you say anything about $$\sigma(2^k n)$$? You know it must be odd if this is quasiperfect...

 

[genii not geniuses]

well sigma(2^k*n) is even.

 

[shmoe]

well sigma(2^k*n) is even.

What makes you say that?

 

[genii not geniuses]

Well, isn't the sum of the divisors of an even number always even?

 

[shmoe]

Well, isn't the sum of the divisors of an even number always even?

No...$$\sigma(2)=3$$, $$\sigma(18)=\sigma(2)\sigma(9)=3\times 13=39$$

 

[genii not geniuses]

Ohhhhhh... I'm used to not counting the number itself, which is why I keep getting confused.

 

[genii not geniuses]

sigma(2^k * n) = odd

 

[shmoe]

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?

 

[genii not geniuses]

No, explain please!

 

[shmoe]

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.

 

[genii not geniuses]

ok, thanks!