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I was going through a simple high school level mathematics book and got to the following question:

n

^{2}- n + 41 is a prime for all positive integers n.

You're supposed to find a counter-example and prove the statement false.

You could of course sit and enter different values for n until you get a composite number and then use that value of n as the counter-example.

But is there a way to find some pattern or rule for prime or composite numbers so that you don't have to do the work manually? This is probably a trivial question but I got curious. Thank you!