Earlier quoted context omitted.
I didn't solve the problems using existing code either. All three are entirely solvable by pencil and paper (or just inspection in the first case).
How do you go about solving #2 on pen and paper? Well, calculating the Fibonacci sequence, ok. The sum of prime divisors of X+1, I guess... though it gets slightly harder for me after dividing by 2, 3 and 5. But how do you know for sure that X is prime? Based on bd's link, I'll just assume that you have super powers. (or I'm approaching the problem the wrong way) edit: I was starting to think about what to write for…
That 317811 is divisible by 3 is obvious upon adding its digits together.
Once you've verified that 514229 isn't divisible by a handful of small primes, you don't try to prove primality. Instead factor 514230 and stick it in. After dividing by 2, 3, and 5, you've got 17141. It doesn't take that long to run through 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61. At which point you have an answer you can try plugging in to discover it is correct. Which is a sequence cperciva pretty definitely has memorized. (I do.) Or alternately cperciva may have better factorization techniques memorized.
If you already have a routine around for factoring (I do), then this takes longer than coding it. But I could do it with paper and pencil pretty easily.