Question

Prove that there are no positive perfect cubes less than 1000 that are the sum of the cubes of two positive integers.

Short answer

There are no positive perfect cubes less than 1000, which can be the addition of cubes of two positive integers.

Step-by-step solution

Introduction

To prove that there are no positive perfect cubes less than 1000 than that are addition of cubes of two positive integers.

The cubes for consideration

The method of exhaustion can be used to obtain the result. A perfect cube is the cube of another number that is an integer.

The cubes that satisfy the criteria here are the cubes of 1 to 9:

1,8,27,64,125,216,343,512,729

It is required to write a perfect cube as the addition of two other cubes that are also perfect.

Mathematically,

x³=y³+z³,z³<x³

Addition for first, second and third cube

It has to be shown here that when the addition is taken for any two numbers in this, it will not be possible to write as an addition of two cubes that are smaller as there are no positive perfect cubes. The first perfect cube one, cannot be written as the addition of two two positive smaller values. The same also applies to to the second perfect cube eight.

Considering the third perfect cube 27, the addition is shown below:

Mathematically,

1+1=2#27

1+8=9#27

8+8=16#27

Addition for fourth and fifth perfect cube

For the fourth perfect cube of 64:

1+27=28≠ 64

8+27=35≠ 64

27+27=54≠64

For the fifth perfect cube 125:

1+64=65≠ 125

8+64=72≠ 125

27+64=91≠ 125

64+64=128≠ 125

Addition for sixth and seventh perfect cube

The next perfect cube 216:

1+125=126≠ 216

8+125=133≠ 216

27+125=152≠ 216

64+125=250≠ 216

125+125=250≠ 216

The seventh perfect cube of 343:

1+216=217≠ 343

8+216=224≠ 343

27+216=243≠ 343

64+216=280≠ 343

125+216=341≠ 343

216+216=432≠ 343

Addition for eighth and ninth perfect cube

The next perfect cube is 512:

1+343=344≠ 512

8+343=224≠ 512

27+343=243≠ 512

64+343=280≠ 512

125+343=341≠ 512

216+343=432≠ 512

343+343=686≠ 512

Finally checking for the last cube 729:

1+512=513≠ 729

8+512=520≠ 729

27+512=539≠ 729

64+512=576≠ 729

125+512=637≠ 729

216+512=728≠ 729

343+512=855≠ 729

512+512=1024≠ 729

It can be seen that none of the addition taken yields value more than 1000 which proves that there are no positive perfect cubes less than 1000, which can be the addition of cubes of two positive integers.

Thus, the required result is found.