Factorise the following trapdoor functions (these small numbers are not good candidates for the trapdoor
functions) into two primes:
(a) 411 (b) 2419 (c) 17 947
(a) Since 411 is not too large, we can test primes up to $\sqrt{411}$.
We find that 3 is a factor, leaving, 137. Testing primes up to $\sqrt{137}$ tells us 137 is prime.
So $411 = 3 \times 137$.
(b) $\lceil \sqrt{2419} \rceil = 50$, and $50^2 - 2419 = 9^2$, and so $2419 = (50-9)(50+9) = 41 \times 59$.
So $2419 = 41 \times 59$.
(c) $\lceil \sqrt{17947} \rceil = 134$ and $134^2 - 17947 = 3^2$, and so $17947 = (134+3)(134-3) = 131 \times 137$.
So $17947 = 131 \times 137$.