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# User:Paul D. Beale

Paul Beale is a Professor of Physics, and Chair of the Department of Physics at the University of Colorado Boulder. He earned a B.S. in Physics with highest honors from the University of North Carolina Chapel Hill in 1977, and a Ph.D. in Physics from Cornell University in 1982. From 1982-1984 he was a postdoctoral research associate at the Department of Theoretical Physics at Oxford University. He joined the Department of Physics at the University of Colorado Boulder in 1984.

Beale's research field is theoretical condensed matter physics, specializing in statistical mechanics. He has made contributions to critical and multicritical phenomena, finite-size scaling, breakdown and fracture properties of random systems, ferroelectric switching, an exact solution of the Ising model on finite lattices, liquid-solid phase transitions, and phase transitions in liquid crystals. He is currnely developing and studying a new class of pseudorandom number generators based on exponential ciphers, and studying phases and phase transitions in layered arrays of electric molecular dipoles. Beale is co-author with Raj K. Pathria of Statistical Mechanics, third edition, (Academic, New York, 2011), the leading graduate physics textbook in the field. Beale added approximately one-hundred pages of original material for the third edition, including new sections and chapters on Bose-Einstein condensation, phase transitions, nonequilibrium statistical mechanics, the thermodynamics of the early universe, and computer simulation methods in statistical physics.

Beale became interested in the integer sequences project as he was developing a new class of pseudorandom number generators based on the Pohlig-Hellman exponential cipher. The implementation uses 32-bit or 64-bit safe primes, i.e. primes for which n and (n-1)/2 are both prime. (This makes (n-1)/2 a Sophie Germaine prime.) There are 3,060,794 safe primes between 2^31 and 2^32. Each safe prime in that range can be used to create a fast, high-quality and independent 32-bit pseudorandom number generator. The 32-bit implementation of the algorithm has millions of possible instances, each with an independent period in excess of 10^18. Sixty-four bit implementations have more that 10^15 instances with periods greater than 10^37.