_{1}

^{*}

For the cube root of a positive integer, a direct method to determine the floor of integer combination of the cube root and its square is given.

The continued fraction expansion of

Because

hence

When one calculates the continued fraction expansion of

Let N be a positive integer and not a cube. Denote

Then these numbers are satisfied the identity:

We achieve two interesting properties on

Theorem 1. If

We consider two cases respectively.

1) If

a) If

since every term in the last expression is nonnegative.

b) If

The above inequality holds because

2) If

a) If

We also have

Hence by the identity (1)

b) If

So we show that

Remark 1. The result is very amazing. Because the quotient ring

But it is surprising that the number

Theorem 2. If

That is to say,

According to Theorem 1,

Hence

So

The proof is completed.

Remark 2. Applying the Theorem 2, we can design an algorithm to calculate the continued fraction expansion of the cube root

The authors wish to thank Prof. Xiangqin Meng for her some helpful advices.

ZhongguoZhou, (2015) Integer Part of Cube Root and Its Combination. Advances in Pure Mathematics,05,774-776. doi: 10.4236/apm.2015.513071