On 8/1/06, Chris Landauer <cal@rush.aero.org> wrote:

hihi, all -

rich asked about consecutive primes p, q, r with

p*q*r = n^3 + 1

for some integer n - i thought i'd take a crack at it without requiring consecutive, and found prety easily

211 * 227 * 241 = 226 ^ 3 + 1 = 11543177,

The sequence of n such that n^3 + 1 = p*q*r with p,q,r distinct primes starts out: 9, 10, 12, 13, 21, 25, 30, 34, 36, 40, 46, 52, 66, 76, 81, 90, 96, 118, 126, 130, 132, 142, 144, 154, 165, 172, 177, 180, 193, 196, 198, 204, 216, 226, 228, 238, 240, 246, 250, 256, 262, 268, 273, 282, 294, 312, 333, 336, 345, 346, 366, 370, 372, 378, 393, 400, 406, 408, 420, 436, 438, 442, 457, 462, 466, 477, 478, 496, 501, 508, 513, ... So, as you can see this happens very often. The above sequence is not in OEIS (maybe I'll add it) but the supersequence not requiring p,q,r distinct is A115403. which leadsd me to think that rich's expectation

(that only 7*11*13 = 1001 works) is correct

BTW, it is easy to show that the middle prime is n+1

The middle prime need not be n+1, for example 2*5*73 = 9^3 +1 but 5 does not equal 9+1 more soon,

cal

Jim Buddenhagen