[math-fun] Product of three consecutive primes question

Quick -- factor 999,999 in your head. [solution 31 lines below] If you got 7*11*13*27*37 fairly quickly, then you probably know that 1001 = 7*11*13, which always seemed like an unlikely curiosity to me. Let p < q < r denote any 3 consecutive primes. How many such p,q,r satisfy p*q*r = N^3 + 1 for some integer N ? (Above all, is it finite or infinite?) If finite, can you list all solutions, or at least give an upper bound for N ? Etc.