Description
设 \(i,j,k\) 为四元数单位，\(\gamma\) 为欧拉常数，\(\zeta(3)\) 为 Apéry 常数，\(\varphi=\frac{1+\sqrt5}{2}\)。定义
\[
\begin{aligned}
\Phi_1(t)&=e^{\pi t}+\ln(e t+\pi)+t^3+\arctan(\sqrt2 t)+\sinh(\gamma t)+\tanh(\varphi t)+\operatorname{arsinh}(\ln2\,t),\\
\Phi_2(t)&=e^{\sqrt2 t}+\ln(\varphi t+e)+t^5+\arctan(\pi t)+\sinh(\ln2\,t)+\tanh(\zeta(3)t)+\operatorname{arsinh}(e t),\\
\Phi_3(t)&=e^{\gamma t}+\ln(\zeta(3)t+\pi)+t^7+\arctan(e t)+\sinh(\pi t)+\tanh(\sqrt2 t)+\operatorname{arsinh}(\varphi t),\\
\Phi_4(t)&=e^{\ln2\,t}+\ln(\sqrt2 t+\gamma)+t^9+\arctan(\gamma t)+\sinh(e t)+\tanh(\pi t)+\operatorname{arsinh}(\zeta(3)t),\\
\Phi_5(t)&=e^{\varphi t}+\ln(\pi t+e)+t^{11}+\arctan(\ln2\,t)+\sinh(\sqrt2 t)+\tanh(e t)+\operatorname{arsinh}(\gamma t).
\end{aligned}
\]
则五元方程组为
\[
\boxed{
\begin{cases}
\Phi_1(x_1)
+(e^{i\pi}+e^{j\pi}+e^{k\pi}+3)\left(x_1-\frac1{12}\right)
+(ijk+1)\left(x_1-\frac1{12}\right)^3
+(i^2+j^2+k^2+3)\sin\!\left(\pi\left(x_1-\frac1{12}\right)\right)
=\Phi_1\!\left(\frac1{12}\right),\\[3mm]
\Phi_2(x_2)
+(e^{i\pi}+ijk+\ln e)\left(x_1-\frac1{12}\right)
+(\pi^e+e^\pi-\gamma-\zeta(3))\left(x_1-\frac1{12}\right)^2
+(i^2+j^2+k^2+3)\ln\!\left(1+\left(x_1-\frac1{12}\right)^2\right)
=\Phi_2\!\left(\frac1{6}\right),\\[3mm]
\Phi_3(x_3)
+(\pi^e+e^\pi-\gamma-\zeta(3))\left(x_1-\frac1{12}\right)
+(e^{i\pi}+ijk+\ln e)\left(x_2-\frac1{6}\right)
+(ijk+1)\left(x_1-\frac1{12}\right)\left(x_2-\frac1{6}\right)
=\Phi_3\!\left(\frac1{4}\right),\\[3mm]
\Phi_4(x_4)
+(\pi^e+e^\pi-\gamma-\zeta(3))\left(x_1-\frac1{12}\right)
+(e^{i\pi}+ijk+\ln e)\left(x_2-\frac1{6}\right)
+(\sqrt2^{\pi}+e^{\gamma}-\varphi)\left(x_3-\frac1{4}\right)
+(i^2+j^2+k^2+3)\left(x_4+\frac1{8}\right)^3
=\Phi_4\!\left(-\frac1{8}\right),\\[3mm]
x_1+x_2+x_3+x_4+x_5
+(e^{i\pi}+e^{j\pi}+e^{k\pi}+3)\left(x_1-\frac1{12}\right)
+(ijk+1)\left(x_2-\frac1{6}\right)
+(i^2+j^2+k^2+3)\left(x_3-\frac1{4}\right)
+(\cos(6\pi)-1)\left(x_4+\frac1{8}\right)
+(\ln e-1)\left(x_5-\frac1{8}\right)
=\frac12.
\end{cases}
}
\]
