Let p₁ and p₂ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of p₁ and p₂?

The correct option is: p₁p₂ is not a prime number.

Let us analyze why this statement is always correct for any two arbitrary prime numbers p₁ and p₂, and why the other options are not always true.

1. Understanding Prime Numbers:
A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.

2. Analyzing the Correct Option (p₁p₂ is not a prime number):
Let $n=p_1{p}_2$ be the product of the two prime numbers.
Since p₁ and p₂ are prime numbers, we have $p_1\ge 2$ and $p_2\ge 2$.
Thus, the product $n$ will have at least the following distinct positive divisors:
1, p₁, p₂, and p₁p₂.
Since $p_1\ge 2$, the factor p₁ is strictly greater than 1 and strictly less than the product p₁p₂.
Therefore, the product p₁p₂ has a divisor other than 1 and itself (specifically, it is divisible by both p₁ and p₂).
By definition, any integer greater than 1 that has divisors other than 1 and itself is a composite number, not a prime number.
Hence, p₁p₂ is not a prime number is always true for all prime numbers p₁ and p₂.

3. Why the other options are incorrect (Counterexamples):
Let us test the other statements with specific values of p₁ and p₂ to show they do not hold in all cases:

• p₁ + p₂ is not a prime number: If we choose $p_1=2$ and $p_2=3$, their sum is $2+3=5$, which is a prime number. Thus, this statement is not always true.
• p₁ + p₂ + 1 is a prime number: If we choose $p_1=3$ and $p_2=5$, then $p_1+p_2+1=3+5+1=9$, which is not a prime number (since 9 is divisible by 3). Thus, this statement is not always true.
• p₁p₂ + 1 is a prime number: If we choose $p_1=3$ and $p_2=5$, then $p_1{p}_2+1=15+1=16$, which is not a prime number. Thus, this statement is not always true.