Implementing a Fast Lucas-Lehmer Test on Programmable Graphics Hardware

The Lucas-Lehmer test provides a deterministic algorithm for testing whether, for a prime number p, Mp = 2−1 is also a prime number. The current work demonstrates that this test can be effectively implemented on a parallel graphics processing unit (GPU). The parallelization was achieved by two main parallel methods: (1) fast multiplication using parallel Fast Fourier transforms in extended precision; (2) fast parallel carryaddition for arbitrary-precision numbers. Extended-precision is necessary in the Fourier transforms to allow single-precision graphics hardware to achieve sufficient precision for tests on non-trivial values of Mp. Methods (1) and (2) allow data to to remain on the graphics card throughout the test and minimize runtime costs of bus traffic between the host and GPU. The algorithm has been implemented in the Cg language and tested on several hardware platforms. The current work demonstrates the viability of current and future GPUs for number theoretic computation. [Addenda (2009): While actual implementations of this were not competitive with highly optimized sequential algorithms such as those used by GIMPS, a similar implementation using modern double-precision GPU hardware and CUDA kernels, rather than Cg-shaders, might produce superior runtimes to sequential algorithms.]