Wednesday, 29 October 2025

Exercise (2.2).10

Show that $(2 × 3 × 5 × 7 × 11 × 13) + 1$ is composite.


 $(2 × 3 × 5 × 7 × 11 × 13) + 1 = 30031$. We only need to test for prime factors less than or equal to $\lfloor \sqrt{30031} \rfloor = 173$. 

The form  $(2 × 3 × 5 × 7 × 11 × 13) + 1$ means it has no prime factors 2,3,5,7,11,13, so these don't need to be tested.

This gives us

$ 30031 = 59 \times 509 $