# Are there infinitely many primes of the form $n^2+n+1$?

When $n>3$, $n$ must be $\equiv 0, 2 \pmod 6$

$3$, $7$, $43$, $73$, $157$, $211$, and $421$ are primes, but $343=7^3$ ($n=18$) is not.

Are there infinitely many primes of the form $n^2+n+1$ ?

• What makes you think that the answer to this question is known? – John Gowers Jan 23 '18 at 15:11
• If you'd like to know, an affirmative answer to this problem would lead to a breakthrough – ArtW Jan 23 '18 at 15:13
• by accident, I was observing uniqueness of positional numeral systems which all digits are same (likewise 111,444,666...) – Jay C. Lee Jan 23 '18 at 15:18
• As far as I know there isn't a single example of a non-linear polynomial which is known to take on infinitely many prime values. Of course, there are lots of conjectures regarding this issue, notably The Bunyakovsky conjecture. – lulu Jan 23 '18 at 15:22
• Yes, it is a special case of the Bunyakovsky conjecture, see this similar question. – Dietrich Burde Jan 23 '18 at 15:23