hihi, all - first, apologies to dan, since i didn't notice that he'd asked the original question second, two more examples (with non-consecutive primes) - 421*443*463 = 442^2 + 1 = 86350889 7*13*19 = 12^3 + 1 = 1729 the first one is part of a family with (p+r)/2 = n, and the second is part of a family with p, q, r in arithmetic progression (i do not know if the families have any other members; i just divided the problem according to the value of (r + p - 2*n) / 2 which is a non-negative integer i have not explored the problem further than this (at least, not fruitfully 8-() more soon, cal