Determine the least positive residue $x$ in $2^{271} \equiv x \pmod {541}$ where 541 is prime.
We use Proposition (7.15) and that prime $541 \equiv -3 \pmod 8$ to conclude that
$$ (\frac{2}{p}) = - 1 $$
By Euler's Criterion, this mean
$$ 2^{\frac{541-1}{2}} \equiv 2^{270} \equiv -1 \pmod {541} $$
Multiplying by 2 we have
$$ 2^{271} \equiv -2 \equiv x \pmod {541} $$
This gives us $x = 539$ as the least positive reside $x$ in $2^{271} \equiv x \pmod {541}$.