Question

If P (n) is the statement "n² − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

Answer 1

$$\begin{gathered} P\left(n\right):n^2-n+41\mathrm{is}\mathrm{prime}. \\ Now, \\ P\left(1\right)=1^2-1+41=41\left(\mathrm{prime}\right) \\ P\left(2\right)=2^2-2+41=4-2+41=43\left(\mathrm{prime}\right) \\ P\left(3\right)=3^2-3+41=9-3+41=47\left(\mathrm{prime}\right) \\ P\left(41\right)={41}^2-41+41=1681(\mathrm{not}\mathrm{prime}) \\ Thus,wecansaythatP\left(1\right),P\left(2\right)andP\left(3\right)\mathrm{are}\mathrm{true},\mathrm{but}P\left(41\right)\mathrm{is}\mathrm{not}\mathrm{true}. \end{gathered}$$