1 (mod p).
This can be rewritten as (a × a-1) + rp = 1,
which implies a fact about the relative primality of a and p
and thus the primality of p.
Proof (part 2) Suppose that every non-zero element of Zp has a multiplicative inverse in Zp.
Then we have
" [a]
Î Zp,
$ [a]-1
Î Zp
such that a × a-1
1 (mod p).
This can be rewritten as (a × a-1) + rp = 1,
which implies a fact about the relative primality of a and p
and thus the primality of p.