Show that if p is a positive integer such that both p and p2

Show that if p is a positive integer such that both p and p^2 + 2 are prime, then p = 3.

Solution

assume p =2

then p=2 , p^2 + 2 = 6 therefore both are not prime

p=4 and p^2 + 2 = 18 therefore both are not prime

p =5 and   p^2 + 2 = 27 therefore both are not prime

p =3 and  p^2 + 2 = 27 therefore prime

so in this way if we substitute any value we will never get p and  p^2 + 2 to be prime , unless p =3

hence by contradiction we proved p =3