?
Let n=1
12 - 1 + 41 = 41 is prime.
n^2 - n + 41 is not prime. so this hypothesis is not correct.
assume n^2 - n + 41 is not prime. there exist n that let n^2 - n + 41 = n^2 so n^2 - n + 41 is not prime.
therefore, when 41-n=0 this equation holds. therefore there exists a n=41 that n^2 - n + 41 is not prime.
Hence n^2 - n + 41 not true for all n is an element of N.
Is this correct?
You can't use induction here. Induction is used to prove statements for all positive integers.
Here, can be prime for some values of (such as n = 1) or not prime (such as n = 41).
To disprove a statement, all you need is just one single counterexample which I provided for you.