I used to answer correspondence for the EFF Cooperative Computing Awards (which are an actual financial prize for finding very large primes) and people who wrot

I used to answer correspondence for the EFF Cooperative Computing Awards (which are an actual financial prize for finding very large primes) and people who wrote in were almost uniformly either kids or people meeting all of the criteria mentioned here. Almost none of them thought it was important to understand any prior results about primality testing before submitting a claim for the prize, while a majority didn't seem comfortable with the idea of a mathematical proof or theorem. In general, they didn't have a sense that some properties are always true, some are never true, and some are sometimes true and sometimes not true, and that mathematical reasoning can often definitively establish which of these categories a particular property is in.