From charlesreid1




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
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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