Does the formula always produce primes?

Statement

Consider the formula $y = x^2 + x + 11$. Is it true that it yields a prime for every integer $x\ge 0$?

Solution

For $x=0,1,\dots,9$ we get $11,\;13,\;17,\;23,\;31,\;41,\;53,\;67,\;83,\;101,$ all prime. But the property does not always hold: for $x=10$ $y = 100 + 10 + 11 = 121 = 11^2,$ which is composite. This single counterexample shows the formula does not always produce primes.