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How can i prove the statement that if the difference of cubes of two consecutive integers is an integral power of 2, then the integer with power 2 can be written as the sum of squares of two different integers.

For example:

$$8^3 - 7^3 = 13^2 = 12^2 + 5^2$$

Any help appreciated.

Thanks.

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This is true of any odd square... See en.wikipedia.org/wiki/… –  Brandon Carter Jun 21 '13 at 20:58
    
@BrandonCarter i want to know how can i prove this. –  Shobhit Jun 21 '13 at 21:01
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2 Answers 2

up vote 1 down vote accepted

We can see that $$n^3-(n-1)^3=3n^2-3n+1.$$ By looking at congruences modulo 4, we see that $$n^3-(n-1)^3=1\pmod 4.$$ Recalling that you assumed that the difference was a square, you can apply Fermat's theorem cited above to conclude.

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Thanks for that. –  Shobhit Jun 21 '13 at 21:40
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Im pretty sure youve heard of an identity $$ a^3-b^3=(a-b)*(a^2 + b^2 + ab) $$
If the above is a perfect square then it must be of form $a+b=n$ and $a^2 + b^2 + ab = n x^2$.

Now you can proceed. I think this method is good for an approach.

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