Prove that there is a quadratic \(f(n) = n^2 + bn + c\), with positive integers coefficients b, c, such that \(f(n)\) is composite (i.e, not prime) for all positive integers n, or else prove that the statement is false.
== Attempt ==
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\[
\begin{aligned}
f(n) & = n^2 + bn + c \\
& = (n^2 + \frac{b}{2})^2 - \frac{b^2}{4} + c
\end{aligned}
\]
|
This means when
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\[
\begin{aligned}
c & = \frac{b^2}{4}
\end{aligned}
\]
|
Then,
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\[
\begin{aligned}
f(n) & = (n^2 + \frac{b}{2})^2
\end{aligned}
\]
|
which is composite !
# posted by rot13(Unafba Pune) @ 11:26 PM
