Answer
$1^{2}-1+11=11$
$2^{2}-2+11=13$
$3^{2}-3+11=17$
$4^{2}-4+11=23$
$5^{2}-5+11=31$
$6^{2}-6+11=41$
$7^{2}-7+11=53$
$8^{2}-8+11=67$
$9^{2}-9+11=83$
$10^{2}-10+11=101$
Work Step by Step
Although inelegant, the method of exhaustion is probably the simplest way to prove a conjecture such as this one. See example 4.1.5 for more on this method of proof.
To show that each of the results ($11$, $13$, $17$, etc.) is in fact prime, we would have to show that none of them are composite, which is generally a difficult task. However, since these primes are all fairly small, we can get away with calling their primality common knowledge.
