Prime Generating Polynomials

Simplest Prime Generating Polynomials

The most famous, due to Euler (1772):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^2 + n + 41 \qquad 0 \leq n \leq 39 }

Slightly modified by Legendre (1798):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^2 - n + 41 \qquad 1 \leq n \leq 40 }

Another simple one due to Legendre was:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 n^2 + 29 \qquad 0 \leq n \leq 28 }

and yet another by Lengendre:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3n^2 + 39n + 37 \qquad 0 \leq n \leq 17 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4n^2 + 4n + 59 \qquad 0 \leq n \leq 13 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2n^2 + 11 \qquad 0 \leq n \leq 10 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 7n^2 - 371n + 4871 \qquad 0 \leq n \leq 23 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 6n^2 - 342n + 4903 \qquad 0 \leq n \leq 57 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 8n^2 - 488n + 7243 \qquad 0 \leq n \leq 61 }

Degree 3

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3n^3 - 183n^2 + 3318n - 18757 \qquad 0 \leq n \leq 46 }

Degree 4

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^4 - 97n^3 + 3294n^2 - 45458n + 21358 \qquad 0 \leq n \leq 49 }