Difference of Two Cubes

Date: 05/24/2001 at 19:37:03
From: Jackie
Subject: A problem in class we have to do

The difference of two cubes is 56,765. What two positive integers 
satisfy this condition?

I have tried the formula (_ ^3-56765)^(1/3). In the blank I put 64 and 
got the integer 59, so I know that is one number, but I don't know 
what to do next. I did most of it by the guess-and-check method. 
Please help.

Date: 05/30/2001 at 13:41:44
From: Doctor Toughy
Subject: Re: A problem in class we have to do

Hi Jackie,

Thanks for writing to Dr. Math.

It looks like you solved the problem. Good job. One cube equals 64^3
and the other cube equals 59^3.

In the future, whenever you're given a problem such as this one where 
there is no clear formula for solving it and the numbers used are 
large, it is usually a good first step to create a simpler problem 
similar to the one given and then determine how to solve the simpler 
problem. Often if you make a table it's possible to spot patterns 
using small numbers in your table that you can later use to solve your 
more difficult problem.  

Suppose we had the following problem:

   The difference of two cubes is 279.  What two positive integers 
   satisfy this condition?

Let a be the larger integer and b the smaller. The equation for this 
would be: 

    a^3 - b^3 = 279

To solve the problem, we create a table using possible values for a 
and b. In the first column we have the value a. In the second column 
we have a^3 - b^3, where b equals a-1. In the third column we have 
a^3 - b^3, where b equals a-2.

In the third column a^3 - b^3 always equals

   a  |  a^3 - b^3        | a^3 - b^3          | a^3 - b^3
      | (where b = a-1)   | (where b = a-2)    | (where b = a-3)
  ____|________________________________________________________      
   1  |   1               |                    |
      |                   |                    |
   2  |   7               |  8                 |
      |                   |                    |
   3  |   19              |  26                |   27
      |                   |                    |
   4  |   37              |  56                |   63
      |                   |                    |
   5  |   61              |  98                |  117
      |                   |                    |
   6  |   91              | 152                |  189


One of the  patterns here is that 

   when b = a-1 then a^3 - b^3 = 1 mod 6,
   when b = a-2 then a^3 - b^3 = 2 mod 6, 
   when b = a-3 then a^3 - b^3 = 3 mod 6
  
Now the modular value of 279 is 3 mod 6, so we know that a-b must 
equal 3 mod 6, which can equal 3, 9, 15 ...

Let's continue the table where b = a-3. As the values of a and b get 
larger, do you see a pattern in the increase in the amount of 
difference between a^3 and b^3, where the difference between a and b 
is 3?  
  
The equation for this difference is 

  difference = 9 * ((a-1)^2 -(a-2))

Now you can follow the same process and solve for a difference of 
56,765.

Hope this helps. If you have any more questions about this or any 
other math topics, please write back.

- Doctor Toughy, The Math Forum
  http://mathforum.org/dr.math/