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Question Number 168458 by aaaspots last updated on 11/Apr/22

$$Howtocheckfgisthesmallest$$ $$hIhavenoidea$$ 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\n

Commented byRasheed.Sindhi last updated on 11/Apr/22

$$f\left(n\right)=n^2+n+17$$ $$f\left(17\right)iscertainly\mathit{c}\mathit{o}\mathit{m}\mathit{p}\mathit{o}\mathit{s}\mathit{i}\mathit{t}\mathit{e}:$$ $$f\left(17\right)={17}^2+17+17=17(17+1+1)$$ $$Sotherequiredn⩽17$$ $$\mathcal{T}estforf\left(1\right),f\left(2\right),...,f\left(16\right).\mathcal{T}{hey}^{\prime }re$$ $$allprimes.$$ $$\therefore \mathcal{T}hesmallestnforwhichf\left(n\right)is$$ $$compositeis17.$$ $$\mathcal{D}{on}^{\prime }tknowsimplermethod.$$

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