Trying to understand Gauss's Lemma

[Oxymoron]

1. Let p=1(mod 8). => p=8n+1

exponent = 8n/4 - floor((8n+1)/4) = 4n - 2n = 2n (EVEN)

2. Let p=-1(mod 8) => p=8n-1

exponent = (8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n = 2n-1 (ODD)

3. Let p=3(mod 8) => p=8n+3

exponent = (8n+2)/2 - floor((8n+3)/4) = 4n - 1 - 2n = 2n-1 (ODD)

4. Let p=-3(mod 8) => p=8n-3

exponent = (8n-4)/2 - floor((8n-3)/4) = 4n - 2 - 2n = 2n-2 (EVEN)

 

[Oxymoron]

But shouldn't the first two exponents be EVEN and the last two ODD?

 

[shmoe]

You have floor((8n-1)/4)=2n, this isn't correct.

 

[Oxymoron]

Sorry, where do I have that?

floor((8n-1)/4) = floor(2n-1/4) which is always going to be 1 less than an even number => ODD

n=1 RHS = 1
n=2 RHS = 3
n=3 RHS = 5
etc...

 

[shmoe]

2. Let p=-1(mod 8) => p=8n-1

exponent = (8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n = 2n-1 (ODD)

(8n-2)/2=4n-1 so

(8n-2)/2 - floor((8n-1)/4) = 4n - 1 - 2n

tells me you have

floor((8n-1)/4) = 2n

 

[Oxymoron]

Ah, I see. Well floor((8n-1)/4) is certainly not 2n. ;)

floor((8n-1)/4) = 2n-1 So we should have

4n - 1 - (2n - 1) = 2n => EVEN


And fixing the mistake in the last case yields ODD

 

[Oxymoron]

Well, if this is all correct then I have shown what I needed to show. :)