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S (n) be the sentence

Joystar1977

Active member
Jul 24, 2013
119
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
We have $41^2-41+41=41^2$, which clearly is not prime. So, $S(n)$ is a false sentence.
 
Last edited:

Joystar1977

Active member
Jul 24, 2013
119
Thanks for explaining this Fernando!I really and truly appreciate it.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
I checked on the computer that $S(n)$ is true for all $n<41$.
 

Joystar1977

Active member
Jul 24, 2013
119
Thank you Evgeny. Makarov for double checking this.

Sincerely,

Joystar1977
 

Joystar1977

Active member
Jul 24, 2013
119
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

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
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.
 

Joystar1977

Active member
Jul 24, 2013
119
Thanks for rechecking on this Evgeny.Makarov!