Number Choosing Contest.

It's worth noting that Kaynato was able to tell us that the largest prime broke rule 2, but computation was a limiting factor in determining its status with respect to rules 1 and 3. This suggests that rule 1 is also not an immediately obvious property of a number, whilst 2 probably is.

EDIT: actually since I'm wasting my posts I may as well ask for the gaps at the start of my spreadsheet to be filled:

12

15