Number theory Proof Question

[CalleighMay]

Hey guys,

I'm in the class "Number Theory" at my college. It's WEEK 1 and we are already going over these weird proofs and i am COMPLETELY lost.

The question asks:

"Show that if n is any odd positive integer, and m = (n^2-1)/2, then m^2 + n^2 = (m+1)^2."

My professor as awful and has given us absolutely NO background.
He's given us two proof examples in class, but i don't understand either of them, so they don't help much.

From what i can gather, odd numbers are "2k+1" and even numbers are "2k". I guess this makes sense, but i don't understand how to use it.

So for the problem on hand, i understand i need to show how you get from point A to point B- but the question is how.

We can assume that n is odd. So n=2k+1.
We can also assume that m= (n^2-1)/2
And we need to get these two equations to look like: m^2 + n^2 = (m+1)^2 in the end

Working backwards:

m^2 + n^2 = (m+1)^2
.............= m^2 + 2m + 1 (expanding)
m^2 + n^2 = m^2 + 2m + 1
........n^2 = ........2m + 1 (subtraction)
so n^2 = 2m+1

Working from the given that m = (n^2 - 1) / 2

m = (n^2 - 1) / 2
2m = n^2 - 1 (multiplication)
2m + 1 = n^2 (addition)
n^2 = 2m + 1 (switch things around)

So this gets the "n^2 = 2m + 1" in the final step. But how do i get the "+ m^2" ??

My biggest problem is that i don't know how this proof should "look". I understand the concept, but i am having difficulty getting it into proof form.

So it should look like:

A: Assume n is positive and odd, and assume m = (n^2 - 1) / 2
A1:
A2:
A3:
A4:
A5:
A6:
A7:
A8:
A9:
B: Show that m^2 + n^2 = (m+1)^2

Then we have to write it in paragraph form....

Can anyone help me out with this problem? I just don't get it...

This problem is from the text "Elementary Number Theory" 2/e by Charles Vanden Eynden.
It's Chapter 0 page 9 number 11.

Thanks for any help!!

 

[Deleted member 4993]

Hey guys,

I'm in the class "Number Theory" at my college. It's WEEK 1 and we are already going over these weird proofs and i am COMPLETELY lost.

The question asks:

"Show that if n is any odd positive integer, and m = (n^2-1)/2, then m^2 + n^2 = (m+1)^2."

Let n = 2k+ 1

m = (n^2-1)/2 = 2k^2+2k

m + 1 = 2k^2 + 2k + 1

(m+1)^2 = 4k^4 + 4k^2 + 1 + 8k^3 +4k^2 + 4k = 4k^4 + 8k^3 + 8k^2 + 4k + 1.....................(1)

m^2 + n^2 = 4k^2(k+1)^2 + (2k+1)^2 ? Expand and show that it is equal to above (1).

My professor as awful and has given us absolutely NO background.
He's given us two proof examples in class, but i don't understand either of them, so they don't help much.

From what i can gather, odd numbers are "2k+1" and even numbers are "2k". I guess this makes sense, but i don't understand how to use it.

So for the problem on hand, i understand i need to show how you get from point A to point B- but the question is how.

We can assume that n is odd. So n=2k+1.
We can also assume that m= (n^2-1)/2
And we need to get these two equations to look like: m^2 + n^2 = (m+1)^2 in the end

Working backwards:

m^2 + n^2 = (m+1)^2
.............= m^2 + 2m + 1 (expanding)
m^2 + n^2 = m^2 + 2m + 1
........n^2 = ........2m + 1 (subtraction)
so n^2 = 2m+1

Working from the given that m = (n^2 - 1) / 2

m = (n^2 - 1) / 2
2m = n^2 - 1 (multiplication)
2m + 1 = n^2 (addition)
n^2 = 2m + 1 (switch things around)

So this gets the "n^2 = 2m + 1" in the final step. But how do i get the "+ m^2" ??

My biggest problem is that i don't know how this proof should "look". I understand the concept, but i am having difficulty getting it into proof form.

So it should look like:

A: Assume n is positive and odd, and assume m = (n^2 - 1) / 2
A1:
A2:
A3:
A4:
A5:
A6:
A7:
A8:
A9:
B: Show that m^2 + n^2 = (m+1)^2

Then we have to write it in paragraph form....

Can anyone help me out with this problem? I just don't get it...

This problem is from the text "Elementary Number Theory" 2/e by Charles Vanden Eynden.
It's Chapter 0 page 9 number 11.

Thanks for any help!!

 

[Denis]

Well, if you attend classes dressed like that, no wonder your teacher can't teach properly :idea:

 

[CalleighMay]

OHHHH. Yeah, i was going about this problem the wrong way.

So you plug in (2k+1) into the m = (n^2 - 1)/2 equation to get a new value of m.
Then you find m^2, (m+1), and (m+1)^2.
Then using n = 2k+1 you find n^2.
Then add these equations of m^2 and n^2 and they will be the same as the equation for (m+1)^2.

I get it!! Thanks Subhotosh Khan!!!

I'll work on the proof and putting it into paragraph form. I think i can do it!!

You guys are awesome. I'm the kind of person where i need a little push then i'm good