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Math Help - Prove

  1. #1
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    Prove

    Prove that there is no perfect square such that the left number is and the right number is and other numbers is .
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  2. #2
    Super Member Bacterius's Avatar
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    Re: Prove

    Let's see... 301... 3001...30001, can we describe this class of integers with a formula? Yes!

    Basically you have to prove that 3 \times 10^n + 1 is not a perfect square for all n > 1 (or n > 0, whatever).

    Thus x^2 = 3 \times 10^n + 1 has no solution for x \in \mathbb{N} for all n > 1.

    The above equation is equivalent to x^2 - 1 = 3 \times 10^n

    ... also known as (x - 1)(x + 1) = 3 \times 10^n

    Can you take it from there?
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  3. #3
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    Re: Prove

    Quote Originally Posted by Bacterius View Post
    Let's see... 301... 3001...30001, can we describe this class of integers with a formula? Yes!

    Basically you have to prove that 3 \times 10^n + 1 is not a perfect square for all n > 1 (or n > 0, whatever).

    Thus x^2 = 3 \times 10^n + 1 has no solution for x \in \mathbb{N} for all n > 1.

    The above equation is equivalent to x^2 - 1 = 3 \times 10^n

    ... also known as (x - 1)(x + 1) = 3 \times 10^n

    Can you take it from there?
    Thanks I am not well in number theory.
    How can we prove that 3 \times 10^n + 1 is not a perfect square for all [TEX]n > 1 ?
    In other method can we write 301... 3001...30001 as 8t+k and say because it cannot be written as 8q+1 is not perfect square ?
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