Step 1: Clarify what is being asked.
Two numbers $p_1$ and $p_2$ are given, and both are prime.
We must identify the statement that remains true for every possible choice of these prime numbers.
Step 2: Recall a basic fact about primes.
A prime number has exactly two positive divisors: 1 and itself.
Any number that has more than two divisors is therefore not prime.
Step 3: Test each option one by one.
(A) $p_1 + p_2$ is not a prime number.
This does not always hold.
For example, if $p_1=2$ and $p_2=3$, then:
\[ p_1+p_2 = 2+3 = 5 \]
Since 5 is prime, this statement fails in some cases.
(B) $p_1 p_2$ is not a prime number.
The product of two prime numbers is always composite.
It is divisible by $1$, $p_1$, $p_2$, and $p_1p_2$ itself.
Hence, it can never be prime, regardless of which primes are chosen.
(C) $p_1 + p_2 + 1$ is a prime number.
This is also not guaranteed.
Taking $p_1=2$ and $p_2=3$ gives:
\[ 2+3+1 = 6 \]
which is clearly not prime.
(D) $p_1 p_2 + 1$ is a prime number.
This fails as well.
For instance, with $p_1=3$ and $p_2=5$:
\[ 3\times5+1 = 16 \]
which is composite.
Step 4: Final conclusion.
The only statement that holds true for all choices of prime numbers is:
\[ \boxed{p_1p_2 \text{ is not a prime number}} \]