Prime Generating Polynomials
From charlesreid1
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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:
$ 2 n^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 $