Animated Solution for Mathematics - Quadratic Equations: Consider the statement : "P(n): $n^2-n+41$ is prime." Then which one of the following is true?

The expression provided is a classic polynomial in number theory:

This expression is famous for generating prime numbers for many consecutive integers. We are tasked with verifying the truth of the proposition $P(n)$ for the specific values $n=3$ and $n=5$.

We begin our investigation by substituting $n=3$ into the expression:

First, we calculate the exponent:

Substituting this back into the expression, we get:

To determine if $47$ is prime, we check for small prime divisors. It is not divisible by $2$, $3$, or $5$. Since $47$ has no divisors other than $1$ and itself, we conclude that $47$ is prime, and therefore $P(3)$ is true.

Next, we turn our attention to $n=5$. We substitute this value into the expression:

Calculating the exponent gives:

The expression simplifies as follows:

We test $61$ for primality by checking small prime factors. It is not divisible by $2$, $3$, or $5$. Testing $7$, we find that $61=7\times 8+5$, which is not divisible. Since we have checked up to the square root of $61$ (approximately $7.8$), we confirm that $61$ is prime, and therefore $P(5)$ is true.

We have successfully evaluated both propositions. We found that $P(3)=47$ and $P(5)=61$, both of which are prime numbers.

Since both statements hold true, the correct conclusion is: (4) Both $P(3)$ and $P(5)$ are true.