Computation of the 100 quadrillionth hexadecimal digit of π on a cluster of Intel Xeon Phi processors

This paper presents the computation of a specific hexadecimal digit of π by using a Bailey-Borwein-Plouffe (BBP)-type formula on a cluster of Intel Xeon Phi processors. The BBP-type formula can be computed using modular exponentiation. We use Montgomery multiplication for the modular multiplication, which is the most time-consuming part of the modular exponentiation. We vectorize multiple modular exponentiations and multiple integer divisions by using Intel Advanced Vector Extensions 512 (Intel AVX-512) instructions. A parallel implementation of the BBP-type formula is presented. The 100 quadrillionth hexadecimal digit of π was computed on a 512-node cluster of Intel Xeon Phi processors with an elapsed time of 641 h 29 min that includes the time required for verification.