Conjecture :
Let $p$ be a prime number such that : $p \equiv 1 \pmod 4$
If multiplicative order of : $b \pmod p$ is $p-1$ then
multiplicative order of : $(p-b) \pmod p$ is $p-1$ .
In other words :
$b^k \equiv 1 \pmod p$ and $(p-b)^k \equiv 1 \pmod p$
for $k=p-1$ , and no $k$ less than this .
Maple code that examines this conjecture :
p := 37;
for b from 2 to p-1 do
i := 0;
for k from 1 to p-1 do
if b^k mod p = 1 then
i := i+1;
end if;
end do;
if i = 1 then
print(b) ;
end if;
end do;
What proof strategy one should use in order to prove this conjecture ?