Posted by **Matt** on Wednesday, July 15, 2009 at 11:27pm.

Prove that a number 10^(3n+1) , where n is a positive integer, cannot be represented as the sum of two cubes of positive integers.

thanx

- Maths -
**MathMate**, Thursday, July 16, 2009 at 6:38am
We will examine the sum of cubes of two numbers, A aand B. Without losing generality, we will further assume that

A=2^{n}X and

B=2^{n+k}Y

where

X is not divisible by 2

n is a positive integer and

k is a non-negative integer.

A^{3}+B^{3}

=(A+B)(A^{2}-AB+B^{2})

=2^{n}(X + 2^{k}Y) 2^{2n}(X^{2} - 2^{k}XY + 2^{2k}Y²)

=2^{3n}(X + 2^{k}Y) (X² - 2^{k}XY + 2^{2k}Y²)

Thus A^{3}+B^{3} has a factor 2^{3n}, but not 2^{3n+1} since X is not divisible by 2.

Since 10^{3n+1} requires a factor of 2^{3n+1}, we conclude that it is not possible that

10^{3n+1}=A^{3}+B^{3}

- Maths -
**Jon Zhan**, Monday, August 3, 2009 at 8:47pm
Nice Answer, But Please Try To Use (Mod)

That Way Is Easier

- Maths -
**Sean**, Monday, August 17, 2009 at 6:01am
Hey, Your ANSWER is corrupt, cause it doesnt really explain anything! Try to make it more clear.

SY

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