The equation m 2 - 4 k = 5 n 2 and unique representations of positive integers
Clark H. Kimberling · The Fibonacci Quarterly · 2007
If n is a positive integer, there exists one and only one pair (j, k) of positive integers such that (j + k + 1)2 − 4k = 5n2. The resulting unique representation of n can be used to generate both the Wythoff difference array and the Fraenkel array. It also provides the solution of the complementary equation b(n) = a(jn) + kn in all cases in which a and b are a pair of Beatty sequences and a(n) is of the form [rn] for r an irrational number in the field Q(√5).