The 'solution' must of course be an integer and that happens only if ...
Kind regards
Hi all, messed up using Latex at first. Okay, got two problems on quadratic reciprocity. I've worked up something of a proof for each, but I feel very uncertain. Thought maybe someone could give me some pointers.
Okay, simple one first:
Suppose p is a prime such that , and let a be a QR modulo p.
Show that is a solution to
My proof:
a - QR mod p the Legendre symbol
,
so clearly .
Multiplying both sides by a yields
.
But if ,
we have that .
We know that is congruent to a modulo p, so
is a solution to the congruence.
My trouble is that this seems too easy, and not once did I apply .
Any pointers? Thanks in advance.