I read in a Book written by Raymond A. Barnett and Micheal R. Ziegler the way to prove conjectures for infinite members of a given set and that is, Mathematical Induction. When I read Induction, I found it very interesting. The way to prove propositions using induction theorem looks like this:
Step 1: Base case i.e. to prove for 1.
Step :2 Inductive step i.e. to assume that proposition is true for any unknown number (k) and then prove for k+1
Then on other page I found an conjecture given by Euler i.e. for each positive integer n, the number n^2 - n + 41 is a prime. When I tried it to prove by Induction theorem stated above, it resulted it to be true for each positive integer however Euler's proposition failed when I put n=41. Is there any other method to check for the Euler's proposition; that could prove it wrong for all n?