# Thread: Here's the proof what's the question!

1. ## Here's the proof what's the question!

This is a 4 part question on my tutorial and if you prove part d) the rest follow, except there's no part d) in the question! I can prove a), b) and c) without using whatever d) is but just thought I'd see if you guys know what this proof is actually proving!

Q. Let p be an odd prime, and a, b be integers not divisible by p. Prove that
a) a \equiv b (mod p) implies that $\displaystyle \bigg{(}\frac{a}{p} \bigg{)} = \bigg{(}\frac{b}{p} \bigg{)}$

b) $\displaystyle \bigg{(}\frac{ab}{p} \bigg{)} = \bigg{(}\frac{a}{p} \bigg{)}\bigg{(}\frac{a}{p} \bigg{)}$.

c) $\displaystyle \bigg{(}\frac{a^2}{p} \bigg{)}=1$; $\displaystyle \bigg{(}\frac{a^2b}{p} \bigg{)} = \bigg{(}\frac{b}{p} \bigg{)}$.

(I though perhaps part d) was the second equality in part c) but I don't think the proof matches up.

PROOF.

a), b) and c) follow from d) so we should prove that first!
d) Take a primitive root g, with $\displaystyle a \equiv g^k$ mod $\displaystyle p$.

Then $\displaystyle a^{\tfrac{p-1}{2}} = g^{\tfrac{(p-1)k}{2}}$.

As $\displaystyle g^{\tfrac{(p-1)}{2}} \equiv -1$ mod $\displaystyle p$ we see that...

$\displaystyle g^{\tfrac{(p-1)k}{2}} \equiv 1$ mod $\displaystyle p$ if $\displaystyle k$ is even (i.e if a is a quadratic residue),

while $\displaystyle g^{\tfrac{(p-1)k}{2}} \equiv -1$ mod $\displaystyle p$ if $\displaystyle k$ is odd (i.e a non quadratic residue). Hence result.

I was thinking it was probably something like show that if g is a primitive root and k is even then,

$\displaystyle \bigg{(}\frac{g^k}{p} \bigg{)} = 1$. and if k is odd then ... = -1.

2. Here's part a. The rest follow in a similar fashion.

$\displaystyle a\equiv b\bmod{p} \implies a^{(p-1)/2}\equiv b^{(p-1)/2}\bmod{p} \implies \left(\frac ap\right) = \left(\frac bp\right)$

3. Originally Posted by chiph588@
Here's part a. The rest follow in a similar fashion.

$\displaystyle a\equiv b\bmod{p} \implies a^{(p-1)/2}\equiv b^{(p-1)/2}\bmod{p} \implies \left(\frac ap\right) = \left(\frac bp\right)$
Lol I think you misread the question...

I can prove a), b) and c) fine. My question was about this mystery part d) that only has a proof but the actual question was not on the tutorial.
Basically, I've been given the proof, can you work out what the question was? My lecturers away so I can't ask him right now.

I only included a,b and c so folk can see the whole question and what part d would imply.

4. Originally Posted by Deadstar
Lol I think you misread the question...

I can prove a), b) and c) fine. My question was about this mystery part d) that only has a proof but the actual question was not on the tutorial.
Basically, I've been given the proof, can you work out what the question was? My lecturers away so I can't ask him right now.

I only included a,b and c so folk can see the whole question and what part d would imply.
Probably too hard to deduce part d.

5. Originally Posted by chiph588@
Probably too hard to deduce part d.
Ah well thought I'd post it up to see...