Tuesday, 30 December 2025

Exercise (4.1).6

Show that $2^{2046} ≡ 1 \pmod {2047}$. Check whether 2047 is prime.


We start with $2^{11} = 2048$, and so

$ 2^{11} \equiv 1 \pmod {2047} $

$  2^{2046} \equiv (2^{11})^{186} \equiv 1 \pmod {2047}$.


2047 is not prime since $2047 = 23 \times 89$.