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Prove that the cube of any integer can be written as the difference of two squares

Chapter 1, Problem 1.4

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QUESTION:

Prove that the cube of any integer can be written as the difference of two squares. [Hint:Notice thatn3 = (13 + 23 + + n3) - (13 + 23 + + (n - 1)3). ]

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QUESTION:

Prove that the cube of any integer can be written as the difference of two squares. [Hint:Notice thatn3 = (13 + 23 + + n3) - (13 + 23 + + (n - 1)3). ]

ANSWER:

Step 1 of 4

Let us consider a is any integer.

Claim that  , where p and q are any integers.

Consider  and add, subtract the number  in we get

 

As we know that,

By using above formula calculate, we have

 

Similarly, calculate

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