How would we design a fast circuit to solve LCS35?
Turns out the LCS35 problem was first solved on April 15, 2019 by Bernard Fabrot. It took 3.5 years of runtime on a single core of a standard modern CPU, using GMP.

Independently, the Cryptophage project solved LCS35 in less than 2 months, starting mid March 2019, using an FPGA-based system that performs modular squaring with extremely low latency.