Solutions in $\mathbb{Z}[i]$ of $A^5+B^5=C^5\pm 1$

We build solutions in $\mathbb{Z}[i]$ of $A^5+B^5=C^5\pm 1$ based on a method developed by Michael D. Hirschhorn in his article “An Amazing Identity of Ramanujan”, from Mathematics Magazine, Vol. 68, No3 (June 1995).