Is the Induction Hypothesis n^2 - n + 41 Prime for All Natural Numbers?

[mbcsantin]

Homework Statement

For all n is an element of N, n² - n + 41 is prime
N= natural numbers

Homework Equations

None.

The Attempt at a Solution

Let n=1
1² - 1 + 41 = 41 is prime.

n² - n + 41 is not prime. so this hypothesis is not correct.
assume n² - n + 41 is not prime. there exist n that let n² - n + 41 = n² so n² - n + 41 is not prime.
therefore, when 41-n=0 this equation holds. therefore there exists a n=41 that n² - n + 41 is not prime.
Hence n² - n + 41 not true for all n is an element of N.

 

[mutton]

You have the right idea, that when n = 41, n^2 - n + 41 is not prime.

Your proof doesn't make sense though. You can't begin by assuming the intended conclusion. It's not clear why there exists an n such that n^2 - n + 41 = n^2 and that this number is not prime. Work on articulating what you mean.

 

[HallsofIvy]

That's not supposed to be a "proof" of anything. It's a perfectly valid way of finding a counter example to the supposed theorem. The "proof" that the theorem is false is simply asserting that if n= 41, n^2- n+ 41= 41^2 is NOT prime. How you got to n= 41 is not relevant to the proof.