Question: Find the smallest positive integer $n$ for which $f(n)=n^2+n+17$ is composite.
Now I know that $f(17)$ is composite as $17^2+17+17= 17(17+2)$ and is divisible by $17$, So either $17$ is the smallest positive integer or there exists an integer $x<17$ which makes $f(x)$ to be composite. I tried for all integer from $1$ to $16$ and found $f(16)=289=17^2$ but I want to know is there any general method to solve such type questions.
I tried getting it on internet and found two answers all of which use the same method (trying for all numbers until $16$.)
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HARSH KUMAR
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Duplicate has been found easily with approach0. – Martin Brandenburg Jan 25 '25 at 14:17
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1@MartinBrandenburg Thanks a lot for telling me about this site, I will always be careful to check for duplicates next time before posting. – HARSH KUMAR Jan 25 '25 at 14:21
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This is an instance of Euler's prime producing polynomials. $\ \ $ – Bill Dubuque Jan 25 '25 at 18:43