its actually insufficient..
use 41 and you find that p is not prime..
so I was wondering when to stop pluggin in
You cannot plug in to prove something. You need to think of the logic why something is true or not true.
Here: p=n^2 -n +41
You want to figure out whether p is prime. Put n = 1, p = 41 (prime)
Put n = 2, p = 43 (prime)
You see there is a pattern. p is prime in these cases.
So now think, will p always be prime?
A prime number has only two factors: 1 and itself. If p can be split into two factors other than 1 and itself, it means it is not prime. (it is much harder to prove that p is prime than to prove that p is not prime). Try to look for a case where p may not be prime.
p=n^2 -n +41 = n(n - 1) + 41
Can you split p into two factors (such that one of them is not 1)? You can if you are able to take something common from n(n-1) and 41. When is this possible?
When n = 41, n = 42 etc
41*40 + 41 = p
p = 41^2
42*41 + 41 = p
p = 41*43
82*81 + 41 = p
p = 41*163
In these cases and many more such cases, p is not prime.
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