Cardano Triplets (noch nicht übersetzt)

A triplet of positive integers (a,b,c) is called a Cardano Triplet if it satisfies the condition:

$$\sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a - b \sqrt{c}} = 1$$

For example, (2,1,5) is a Cardano Triplet.

There exist 149 Cardano Triplets for which a+b+c ≤ 1000.

Find how many Cardano Triplets exist such that a+b+c ≤ 110,000,000.