Entry 80

Level 10. Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) and the McKay-Thompson series of class 10A for the Monster. 

$$j_{10A}(\tau) = \left(\left(\frac{d_2\,d_5}{d_1\,d_{10}}\right)^3-\left(\frac{d_1\,d_{10}}{d_2\,d_5}\right)^3\right)^2$$ Examples. We select \(d=20m\) with class number \(h(-d)=4\) and find odd \(m=17\) as well as $$m=6, 14, 26, 38$$ such that the following are well-behaved integers $$\begin{align}j_{10A}\Big(\tfrac{1+\sqrt{-17/5}}{2}\Big) &= -18^2\\ j_{10A}\big(\tfrac12\sqrt{-6/5}\big) &= 6^2\\ j_{10A}\big(\tfrac12\sqrt{-14/5}\big) &= 14^2\\ j_{10A}\big(\tfrac12\sqrt{-26/5}\big) &= 36^2\\ j_{10A}\big(\tfrac12\sqrt{-38/5}\big) &= 76^2 \end{align}$$ Note that the last is responsible for the prime-generating polynomial $$F(n)=10n^2+19$$ which has discriminant \(d = -5\times 38 = -190\) and is prime for \(19\) consecutive values \(n=0 - 18\).