On the Representations of xy + yz + ZX

In view of this representation, Crandall asked what integers are of the form of xy + yz + xz with x, y, z ≥ 1 and he made a conjecture that every odd integer greater than one can be written as xy + yz + xz. In this manuscript, we shall show that Crandall’s conjecture is indeed true. In fact, we are able to show Theorem 1.1. There are at most 19 integers which are not in the form of xy + yz + xz with x, y, z ≥ 1. The only such non-square-free integers are the numbers 4 and 18. The first 16 square-free integers are 1, 2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462. (1.1)