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Prime Factorization
By 2ndSequence, history, 4 years ago, In English

Is it possible to do prime factorization for (A + B) before doing the summation? I mean lets suppose that we want to prime factorize A^n + B^n and 1 <= A, B, N <= 1e12 .... how can i do so? or it's impossible and it's a stupid question?..

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4 years ago, # |
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The following is a related question: If $$$A$$$ and $$$B$$$ are two positive co-prime integers, i.e.

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, does their sum

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have any prime factor that appears in $$$A$$$ or $$$B$$$?

Another related question: Is it possible to express a prime number $$$p$$$ as

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, where $$$A$$$ and $$$B$$$ are positive integers that are NOT co-prime?
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    4 years ago, # ^ |
    ← Rev. 2   Vote: I like it +13 Vote: I do not like it

    The answer is negative for both questions:

    $$$A+B$$$ can't have any prime factors in common with $$$A$$$ or $$$B$$$. Because by the Euclidian Algorithm,

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    .

    For the second one, if

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    ,

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    . So

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    , where

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    and

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    . This contradicts the assumption of $$$p$$$ being prime.
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