When do we 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.

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

Thanks Karishma, but I didn't understand the above..

specifically

41*40 + 41 = p
p = 41^2 and

You can if you are able to take something common from n(n-1) and 41
Please help. .

A prime number has no factors other than 1 and itself.
If I say that x = a*b and a and b are positive non-1 integers, can I say that x is not prime? Sure. x has two factors a and b which are not 1 (and hence not x either).
What we are trying to do here is trying to find whether there is a similar pair of factors that p has.

p = n(n-1) + 41
p can have two factors if we can express p like this: p = (..)*(...)
To do that, we will need to take something common from n(n-1) and 41. Say if n = 41, then we can take something common
p = 41*40 + 41
p = (41) *(40 + 1)
Notice that p is the product of 2 factors in this case. Neither one of the factors is 1. Hence, p is not prime.

Thanks a lot Karishma, Prime nos seem easy now

Do you have any blogs on veritas website on prime nos?