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Prime Factorization
By 2ndSequence, history, 5 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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5 years ago, # |
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The following is a related question: If $$$A$$$ and $$$B$$$ are two positive co-prime integers, i.e. $$$\gcd(A,B) = 1$$$, does their sum $$$(A+B)$$$ have any prime factor that appears in $$$A$$$ or $$$B$$$?

Another related question: Is it possible to express a prime number $$$p$$$ as $$$p = A+B$$$, where $$$A$$$ and $$$B$$$ are positive integers that are NOT co-prime?

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    5 years ago, # ^ |
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    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, $$$gcd(A, A+B) = gcd(A, B) = 1 = gcd(A+B, B)$$$.

    For the second one, if $$$g = gcd(A, B) > 1$$$, $$$A = gA'$$$and $$$B = gB'$$$. So $$$p = g(A'+B')$$$, where $$$g > 1$$$ and $$$A'+B' > 1$$$. This contradicts the assumption of $$$p$$$ being prime.

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