Divisibility on the sequence of perfect squares minus one: The gap principle.

In this paper, we consider a gap principle when a 2 − 1 | b 2 − 1 | c 2 − 1 with 1 < a < b < c . As a byproduct, we are led to determine the complete set of pairs of positive integers 1 ≤ u ≤ v ≤ x such that u | v 2 − 1 and v | u 2 − 1 and the diophantine equation u 2 + v 2 − 1 = m u v . We also generalize our main theorems to the polynomial f ( n ) = A ( n + B ) 2 + C . [ABSTRACT FROM AUTHOR]