Here's how you show the hint : by Euler's criterion, . This immediately implies that the Legendre symbol is multiplicative in the top argument; in perticular if neither are quadratic residues then is a quadratic residue.
I am having troubles with this question:
Show that if p is prime and p >= 7, then there are always two consecutive residues of p (Hint: Show that at least one of 2,5,10 is a quadratic residue of p).
I was thinking of using the values of (2/p) and (5/p), but i am not sure where to go from there.
i feel silly, i was trying to use theorems from Gauss' lemma to show the hint
Okay now have i have if 2 is a quadratic residue of p, and 1 is a quadratic residue of every number, then we have 2 consecutive residues. However I'm having trouble with 5 and 10. Would it have to do with the fact that 5 and 10 are adjacent to 4 and 9 respectively?
Perhaps; I haven't really tried to think about the relevance of the hint to the original question.
Let's solve it in the most general manner possible. We will be working in and will not bother writing but instead we'll just write . We also take p sufficiently large (which is not quite large at all).
Lemma 1. For some , has a solution with .
Proof : find with (this is where we need large enough). Then is a solution.
Theorem. Every square in is the sum of two nonzero squares.
Proof : Suppose we want to write as a sum of two nonzero squares. We write and then we have, using the lemma above, .
In perticular, if we take then we have ; i.e. there are two quadratic residues differing by 1.
Don't hesitate to ask if you're not convinced.
Sorry for not using the hint
See a more general statement here.
(Set for your particular case, you'd have to check the case p=7 separately though.)
EDIT: To add that in the case p=7, both 1 and 2 ( 3² = 9) are quadratic residues module 7.
I'm assuming that's field theory you did there bruno, and my knowledge of fields is very limited. It looks like a very rigourous proof but my number theory class doesn't use any results from abstract algebra. We're expected to prove this using properties of congruences and residues.