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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

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Matt

  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=2n+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 + 2kY) 22n(X2 - 2kXY + 22kY²)
    =23n(X + 2kY) (X² - 2kXY + 22kY²)
    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

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    MathMate

  2. Nice Answer, But Please Try To Use (Mod)

    That Way Is Easier

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    Jon Zhan

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

    SY

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    Sean

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