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This question is inspired by question A5 from the Putnam Mathematical Competition:

Let $$P_n(x) = 1 + 2x + 3x^2 + \cdots + nx^{n-1}.$$ Prove that polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j \neq k$.

I saw their solutions, and it is interesting but I think a different way is to perhaps use abstract algebra. I saw another question on Math.se: Relatively Prime over C

Is there another way to solve the problem, aside from the solutions they suggestion? In particular, something from number theory or abstract algebra.

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Amad27
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  • An additional solution using $p$-adic valuations is mentioned, this matches your criteria. It is not quite clear what you want. Further, I find it strange that you do (edit: did) not even include the problem itself here (edit: leaving this work for somebody else to do). – quid Dec 28 '14 at 12:14
  • @quid, sorry. I linked it because I was sure I was going to mess something up in typing the problem. Im sorry, i wont do this anymore. – Amad27 Dec 28 '14 at 13:02
  • There's a theorem that say that a polynomial is irreducible if it takes on "too many" prime values. There's probably a theorem that says two polynomials are relatively prime if they take on too many relatively prime values. But it will be an "if", not an "if and only if" result. – Gerry Myerson Jan 02 '15 at 01:22

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