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I have the following question:

Find the largest prime factor of: $17^3 - 5^3$

Without any theory or tricks, I solved this the long way and I got $19$ as the answer. Though I would like to know if there any quick way of solving this. I understand the concepts of exponentiation and prime decomposition, but that doesn't help me here.

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    $x^3-y^3=(x-y)(x^2+xy+y^2)$ – 2017-02-13
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    Do you mean "Find **a** prime factor"? Or "Find the prime factorization"? The first is trivial: both $17^3$ and $5^3$ are odd, so their difference is even, and $2$ is a prime factor. – 2017-02-13
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    @Wolfram, perhaps what's meant is to find the *largest* prime factor. – 2017-02-13

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Use the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$, where $a=17$ and $b=5$.

You will end up with $(12)(399)$.

Notice that $399=400-1=20^2-1^2$. Use the difference of two squares identity.

So $399=(20+1)(20-1)=21*19$.

Therefore the expression is equal to $12*21*19$. It is easy to see that the prime factors are $2, 3, 7$ and $19$.

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    ... then, notice that $21=5^2-2^2=(5-2)(5+2)\quad \ddot\smile$. – 2017-02-13