Heegner Number - Other Heegner Numbers

Other Heegner Numbers

For the four largest Heegner numbers, the approximations one obtains are as follows.

$\begin{align} e^{\pi \sqrt{19}} &\approx 96^3+744-0.22\\ e^{\pi \sqrt{43}} &\approx 960^3+744-0.00022\\ e^{\pi \sqrt{67}} &\approx 5280^3+744-0.0000013\\ e^{\pi \sqrt{163}} &\approx 640320^3+744-0.00000000000075 \end{align}$

Alternatively,

$\begin{align} e^{\pi \sqrt{19}} &\approx 12^3(3^2-1)^3+744-0.22\\ e^{\pi \sqrt{43}} &\approx 12^3(9^2-1)^3+744-0.00022\\ e^{\pi \sqrt{67}} &\approx 12^3(21^2-1)^3+744-0.0000013\\ e^{\pi \sqrt{163}} &\approx 12^3(231^2-1)^3+744-0.00000000000075 \end{align}$

where the reason for the squares is due to certain Eisenstein series. For Heegner numbers, one does not obtain an almost integer; even is not noteworthy. The integer j-invariants are highly factorisable, which follows from the form, and factor as,

$\begin{align} j((1+\sqrt{-19})/2) &= 96^3 =(2^5 \cdot 3)^3\\ j((1+\sqrt{-43})/2) &= 960^3=(2^6 \cdot 3 \cdot 5)^3\\ j((1+\sqrt{-67})/2) & =5280^3=(2^5 \cdot 3 \cdot 5 \cdot 11)^3\\ j((1+\sqrt{-163})/2) &=640320^3=(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29)^3. \end{align}$

These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,

$\begin{align} e^{\pi \sqrt{19}} &\approx x^{24}-24; x^3-2x-2=0\\ e^{\pi \sqrt{43}} &\approx x^{24}-24; x^3-2x^2-2=0\\ e^{\pi \sqrt{67}} &\approx x^{24}-24; x^3-2x^2-2x-2=0\\ e^{\pi \sqrt{163}} &\approx x^{24}-24; x^3-6x^2+4x-2=0 \end{align}$

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. In addition, they can also be closely approximated by algebraic numbers of degree 4,

$\begin{align} e^{\pi \sqrt{19}} &\approx 3^5 \left(3-\sqrt{2(-3+1\sqrt{3\cdot19})} \right)^{-2}-12.00006\dots\\ e^{\pi \sqrt{43}} &\approx 3^5 \left(9-\sqrt{2(-39+7\sqrt{3\cdot43})} \right)^{-2}-12.000000061\dots\\ e^{\pi \sqrt{67}} &\approx 3^5 \left(21-\sqrt{2(-219+31\sqrt{3\cdot67})} \right)^{-2}-12.00000000036\dots\\ e^{\pi \sqrt{163}} &\approx 3^5 \left(231-\sqrt{2(-26679+2413\sqrt{3\cdot163})} \right)^{-2}-12.00000000000000021\dots \end{align}$

Note the reappearance of the integers as well as the fact that,

$\begin{align} &2^6 \cdot 3(-3^2+3 \cdot 19 \cdot 1^2) = 96^2\\ &2^6 \cdot 3(-39^2+3 \cdot 43 \cdot 7^2) = 960^2\\ &2^6 \cdot 3(-219^2+3 \cdot 67 \cdot 31^2) = 5280^2\\ &2^6 \cdot 3(-26679^2+3 \cdot 163 \cdot 2413^2) = 640320^2 \end{align}$

which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,

$\begin{align} e^{\pi \sqrt{19}} &\approx (5x)^3-6.000010\dots\\ e^{\pi \sqrt{43}} &\approx (5x)^3-6.000000010\dots\\ e^{\pi \sqrt{67}} &\approx (5x)^3-6.000000000061\dots\\ e^{\pi \sqrt{163}} &\approx (5x)^3-6.000000000000000034\dots \end{align}$

where the xs are given respectively by the appropriate root of the sextic equations,

$\begin{align} &5x^6-96x^5-10x^3+1=0\\ &5x^6-960x^5-10x^3+1=0\\ &5x^6-5280x^5-10x^3+1=0\\ &5x^6-640320x^5-10x^3+1=0 \end{align}$

with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension (with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let, then,

$\begin{align} e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/24} \eta(\tau)}{\eta(2\tau)} \right)^{24}-24.00000000000000105\dots\\ e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/12} \eta(\tau)}{\eta(3\tau)} \right)^{12}-12.00000000000000021\dots\\ e^{\pi \sqrt{163}} &= \left( \frac{e^{\pi i/6} \eta(\tau)}{\eta(5\tau)} \right)^{6}-6.000000000000000034\dots \end{align}$

where the eta quotients are the algebraic numbers given above.

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