Prime Generating Polynomials

Simplest Prime Generating Polynomials

The most famous, due to Euler (1772):

$n^{2}+n+41\qquad 0\leq n\leq 39$

Slightly modified by Legendre (1798):

$n^{2}-n+41\qquad 1\leq n\leq 40$

Another simple one due to Legendre was:

$2n^{2}+29\qquad 0\leq n\leq 28$

and yet another by Lengendre:

$n^{2}+n+17\qquad 0\leq n\leq 15$

Degree 2

These are listed at Wolfram Mathworld's Prime Generating Polynomials page: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

$3n^{2}+39n+37\qquad 0\leq n\leq 17$

$4n^{2}+4n+59\qquad 0\leq n\leq 13$

$2n^{2}+11\qquad 0\leq n\leq 10$

$7n^{2}-371n+4871\qquad 0\leq n\leq 23$

$6n^{2}-342n+4903\qquad 0\leq n\leq 57$

$8n^{2}-488n+7243\qquad 0\leq n\leq 61$

Degree 3

$3n^{3}-183n^{2}+3318n-18757\qquad 0\leq n\leq 46$

Degree 4

$n^{4}-97n^{3}+3294n^{2}-45458n+21358\qquad 0\leq n\leq 49$