ヘーグナー数
000\,000\,000\,000\,000\,21\dots \end{aligned}}}
括弧内の式を x {\displaystyle x} とおくと(例: x = 3 − 2 ( 1 − 96 / 24 + 1 3 ⋅ 19 ) {\displaystyle x=3-{\sqrt {2(1-96/24+1{\sqrt {3\cdot 19}})}}} )、 x {\displaystyle x} はそれぞれ四次方程式を満たす。
x 4 − 4 ⋅ 3 x 3 + 2 3 ( 96 + 3 ) x 2 − 2 3 ⋅ 3 ( 96 − 6 ) x − 3 = 0 x 4 − 4 ⋅ 9 x 3 + 2 3 ( 960 + 3 ) x 2 − 2 3 ⋅ 9 ( 960 − 6 ) x − 3 = 0 x 4 − 4 ⋅ 21 x 3 + 2 3 ( 5 280 + 3 ) x 2 − 2 3 ⋅ 21 ( 5 280 − 6 ) x − 3 = 0 x 4 − 4 ⋅ 231 x 3 + 2 3 ( 640 320 + 3 ) x 2 − 2 3 ⋅ 231 ( 640 320 − 6 ) x − 3 = 0 {\displaystyle {\begin{aligned}&x^{4}-4\cdot 3x^{3}+{\tfrac {2}{3}}(96+3)x^{2}\qquad \quad -{\tfrac {2}{3}}\cdot 3(96-6)x-3=0\\&x^{4}-4\cdot 9x^{3}+{\tfrac {2}{3}}(960+3)x^{2}\ \ \quad \quad -{\tfrac {2}{3}}\cdot 9(960-6)x-3=0\\&x^{4}-4\cdot 21x^{3}+{\tfrac {2}{3}}(5\,280+3)x^{2}\quad \ \;-{\tfrac {2}{3}}\cdot 21(5\,280-6)x-3=0\\&x^{4}-4\cdot 231x^{3}+{\tfrac {2}{3}}(640\,320+3)x^{2}-{\tfrac {2}{3}}\cdot 231(640\,320-6)x-3=0\\\end{aligned}}}
大きい方から4つのヘーグナー数について、以下の近似が得られる 。
e π 19 ≈ 96 3 + 744 − 0.22 e π 43 ≈ 960 3 + 744 − 0.000 22 e π 67 ≈ 5 280 3 + 744 − 0.000 0013 e π 163 ≈ 640 320 3 + 744 − 0.000 000 000 000 75 {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx 960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx 5\,280^{3}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 640\,320^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}}
あるいは、
e π 19 ≈ 12 3 ( 3 2 − 1 ) 3 + 744 − 0.22 e π 43 ≈ 12 3 ( 9 2 − 1 ) 3 + 744 − 0.000 22 e π 67 ≈ 12 3 ( 21 2 − 1 ) 3 + 744 − 0.000 0013 e π 163 ≈ 12 3 ( 231 2 − 1 ) 3 + 744 − 0.000 000 000 000 75 {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 12^{3}(3^{2}-1)^{3}+744-0.
000 000 061 ... e π 67 ≈ 3 5 ( 21 − 2 ( 1 − 5 280 / 24 + 31 3 ⋅ 67 ) ) − 2 − 12.000 000 000 36 ... e π 163 ≈ 3 5 ( 231 − 2 ( 1 − 640 320 / 24 + 2 413 3 ⋅ 163 ) ) − 2 − 12.000 000 000 000 000 21 ... {\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 3^{5}\left(3-{\sqrt {2(1-96/24+1{\sqrt {3\cdot 19}})}}\right)^{-2}-12.000\,06\dots \\e^{\pi {\sqrt {43}}}&\approx 3^{5}\left(9-{\sqrt {2(1-960/24+7{\sqrt {3\cdot 43}})}}\right)^{-2}-12.000\,000\,061\dots \\e^{\pi {\sqrt {67}}}&\approx 3^{5}\left(21-{\sqrt {2(1-5\,280/24+31{\sqrt {3\cdot 67}})}}\right)^{-2}-12.000\,000\,000\,36\dots \\e^{\pi {\sqrt {163}}}&\approx 3^{5}\left(231-{\sqrt {2(1-640\,320/24+2\,413{\sqrt {3\cdot 163}})}}\right)^{-2}-12.