MHB Determine if "n squared - n + 41 is prime" is True or False

[Joystar77]

Let S (n) be the sentence

n squared - n + 41 is prime for all natural numbers n.

Determine if S (n) is a true or false sentence.

Is this a true sentence? If not, can somebody please explain this to me?

 

[Fernando Revilla]

We have $41^2-41+41=41^2$, which clearly is not prime. So, $S(n)$ is a false sentence.

 

[Joystar77]

Thanks for explaining this Fernando!I really and truly appreciate it.

 

[Evgeny.Makarov]

I checked on the computer that $S(n)$ is true for all $n<41$.

 

[Joystar77]

Thank you Evgeny. Makarov for double checking this.



Joystar1977

 

[Joystar77]

Evgeny.Makarov is this problem done correctly?

Let S (n) be the sentence

n squared - n + 41 is prime for all natural numbers n.

S (41) = 41 squared - 41 + 41 = 41 squared, which clearly is not prime.

 

[Evgeny.Makarov]

Let S (n) be the sentence

n squared - n + 41 is prime for all natural numbers n.

S (41) = 41 squared - 41 + 41 = 41 squared, which clearly is not prime.

First, there may be a typo in the problem statement. It should say either "Let $S(n)$ be '$n^2-n+41$ is prime'" or "Let $S$ be '$n^2-n+41$ is prime for all $n$'". Recall that a proposition is something that can be either true or false. In the first case the truth value of $S(n)$ depends on $n$, and for each concrete $n$, $S(n)$ is a proposition. In the second case the truth value of $S$ does not depend on anything, and $S$ itself is a proposition.

Let's assume we have the first case. Then $S(41)$ is a proposition, i.e., true or false. It is important that, in particular, it cannot equal a number and you can't write that $S(41)=41$. Instead, you should write, "When $n=41$, $n^2-n+41=41^2$, which is not prime; therefore, $S(41)$ is false, which in turn means that "For all $n$, $S(n)$" is also false.

Hint: It is customary to write n^2 for $n^2$ in plain text.

 

[Joystar77]

Thanks for rechecking on this Evgeny.Makarov!