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How-to-check-f-g-is-the-smallest-h-I-have-no-idea-Find-the-smallest-positive-integer-n-for-which-the-function-f-n-n-2-n-17-is-composite-Do-the-same-for-the-functions-g-n-n-2-21n-1-and-h




Question Number 168458 by aaaspots last updated on 11/Apr/22
How to check f g is the smallest  h I have no idea  Find the smallest positive integer n for which the function f(n) = n^2 + n + 17 is composite. Do the same for the functions g(n) = n^2 + 21n + 1 and h(n) = 3n^2 + 3n + 23
$${How}\:{to}\:{check}\:{f}\:{g}\:{is}\:{the}\:{smallest} \\ $$$${h}\:{I}\:{have}\:{no}\:{idea} \\ $$Find the smallest positive integer n for which the function f(n) = n^2 + n + 17 is composite. Do the same for the functions g(n) = n^2 + 21n + 1 and h(n) = 3n^2 + 3n + 23
Commented by Rasheed.Sindhi last updated on 11/Apr/22
f(n)=n^2 +n+17  f(17) is certainly composite:    f(17)=17^2 +17+17=17(17+1+1)  So the required n≤17  Test for f(1),f(2),...,f(16).They′re  all primes.  ∴ The smallest n for which f(n) is  composite is 17.  Don′t know simpler method.
$${f}\left({n}\right)={n}^{\mathrm{2}} +{n}+\mathrm{17} \\ $$$${f}\left(\mathrm{17}\right)\:{is}\:{certainly}\:\boldsymbol{{composite}}: \\ $$$$\:\:{f}\left(\mathrm{17}\right)=\mathrm{17}^{\mathrm{2}} +\mathrm{17}+\mathrm{17}=\mathrm{17}\left(\mathrm{17}+\mathrm{1}+\mathrm{1}\right) \\ $$$${So}\:{the}\:{required}\:{n}\leqslant\mathrm{17} \\ $$$$\mathcal{T}{est}\:{for}\:{f}\left(\mathrm{1}\right),{f}\left(\mathrm{2}\right),…,{f}\left(\mathrm{16}\right).\mathcal{T}{hey}'{re} \\ $$$${all}\:{primes}. \\ $$$$\therefore\:\mathcal{T}{he}\:{smallest}\:{n}\:{for}\:{which}\:{f}\left({n}\right)\:{is} \\ $$$${composite}\:{is}\:\mathrm{17}. \\ $$$$\mathcal{D}{on}'{t}\:{know}\:{simpler}\:{method}. \\ $$

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