Entry 34

Here are some optimum prime-generating polynomials $$\begin{aligned}
P(n) &=n^2+n+41\\
P(n) &=2n^2+29\\
P(n) &=2n^2+2n+19\\
P(n) &=3n^2+3n+23\\
P(n) &=5n^2+5n+13\\
P(n) &=6n^2+6n+31\\
P(n) &=7n^2+7n+17
\end{aligned}$$The first one is the most famous, being Euler's. Expressed as the quadratic \(P(n) = an^2+bn+c\), its discriminant is \(d=b^2-4ac\). Using the values \(a,d\) of the above, one gets
$$\begin{aligned}
&e^{\pi/1\,\sqrt{163}} = 640320^3 +743.9999999999992\dots\\
&e^{\pi/2\,\sqrt{232}} = 396^4 -104.0000001\dots\\
&e^{\pi/2\,\sqrt{148}} = (84\sqrt{2})^4 +103.99997\dots\\
&e^{\pi/3\,\sqrt{267}} = 300^3 + 41.99997\dots\\
&e^{\pi/5\,\sqrt{5\times47}} = (18\sqrt{47})^2 + 15.991\dots\\
&e^{\pi/6\,\sqrt{6\times118}} = 1060^2 + 9.99992\dots\\
&e^{\pi/7\,\sqrt{7\times61}} = (39\sqrt{7})^2 + 9.995\dots
\end{aligned}$$These approximations (actually, the exact values of certain eta quotients) can be used as denominators in pi formulas known as Ramanujan-Sato series to be discussed in Entry 35.