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.

Question

thanx

Answer 1

We will examine the sum of cubes of two numbers, A and B. Without losing generality, we will further assume that
A=2ⁿX and
B=2^n+kY
where
X is not divisible by 2
n is a positive integer and
k is a non-negative integer.

A³+B³
=(A+B)(A²-AB+B²)
=2ⁿ(X + 2^kY) 2²ⁿ(X² - 2^kXY + 2^2kY²)
=2³ⁿ(X + 2^kY) (X² - 2^kXY + 2^2kY²)
Thus A³+B³ has a factor 2³ⁿ, but not 2³ⁿ⁺¹ since X is not divisible by 2.
Since 10³ⁿ⁺¹ requires a factor of 2³ⁿ⁺¹, we conclude that it is not possible that
10³ⁿ⁺¹=A³+B³

Answer 2

Nice Answer, But Please Try To Use (Mod)

That Way Is Easier

Answer 3

Hey, Your ANSWER is corrupt, cause it doesnt really explain anything! Try to make it more clear.

SY