## EXPLORATION 2 Is n2+n+41 Prime for All n ?
1. Plug in the numbers from 1 to 10 . Are the results all prime?
2. Repeat for the numbers from 11 to 20.
3. Repeat for the numbers from 21 to 30 . (Ready to make your conjecture?)
4. What is the smallest value of n for which n2+n+41 is not prime?
There is one situation in which (nonmathematical) induction can constitute a proof. In enumerative induction, one reasons from specific cases to the general principle by considering all possible cases. This is simple enough when proving a theorem like "All one-digit prime numbers are factors of 210," but it can involve some very elegant mathematics when the number of cases is seemingly infinite. Such was the case in the proof of the Four-Color Map Theorem, in which all possible cases were settled with the help of a clever computer program.