Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?

Prime Number Properties Analysis

This solution examines the properties of statements involving two arbitrary prime numbers, $p_1$ and $p_2$. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Evaluating Prime Number Statements

- **Statement 1: $p_1+p_2$ is not a prime number.** - Test Case: Let $p_1=2$ and $p_2=3$. Both are prime. - Sum: $p_1+p_2=2+3=5$. - Result: 5 is a prime number. Therefore, this statement is false as it's not true for all primes.
- **Statement 2: $p_1{p}_2$ is not a prime number.** - Definition: A prime number has only 1 and itself as divisors. - Consider the product $p_1{p}_2$. Since $p_1$ and $p_2$ are primes, they are both greater than 1. - The divisors of $p_1{p}_2$ include $1$, $p_1$, $p_2$, and $p_1{p}_2$. - As $p_1>1$ and $p_2>1$, $p_1$ is a divisor other than 1 and $p_1{p}_2$. Similarly for $p_2$. - Therefore, $p_1{p}_2$ always has more than two divisors, making it a composite number. - Result: This statement is always true.
- **Statement 3: $p_1+p_2+1$ is a prime number.** - Test Case: Let $p_1=2$ and $p_2=3$. - Sum + 1: $p_1+p_2+1=2+3+1=6$. - Result: 6 is not a prime number. Therefore, this statement is false.
- **Statement 4: $p_1{p}_2+1$ is a prime number.** - Test Case: Let $p_1=3$ and $p_2=5$. Both are prime. - Product + 1: $p_1{p}_2+1=(3)(5)+1=15+1=16$. - Result: 16 is not a prime number. Therefore, this statement is false.

Correct Statement Identification

The analysis confirms that only the second statement, "$p_1{p}_2$ is not a prime number", is universally true for any two prime numbers $p_1$ and $p_2$. This is because the product of two primes will always have $p_1$ and $p_2$ as factors, in addition to 1 and the product itself, disqualifying it from being prime.