19. Erin lists all three-digit primes that are 21 less than a square. What is the mean of the numbers in Erin's list?
19. 421 When an odd number is subtracted from an odd square, an even (and hence composite) number is obtained. Similarly, when a multiple of 3 (or 7) is subtracted from a square of a multiple of 3 (or 7), a multiple of 3 (or 7) is obtained which is also composite. Therefore we need only consider three-digit squares that are neither odd nor a multiple of 3 (or 7). Hence the only squares we need to consider are $16^{2}=256,20^{2}=400$, $22^{2}=484$ and $26^{2}=676$ which yield differences of $235,379,463$ and 655 respectively. It is easy to see that 235 and 655 are multiples of 5 and hence composite. Therefore only 379 and 463 remain as possible primes satisfying the given condition. After checking divisibilty by $11,13,17$ and 19 for both, both are indeed seen to be prime and their mean is 421 .