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I have been reading on https://www.pcg-random.org/pdf/hmc-cs-2014-0905.pdf about how to do some random number generation. It involves LCGs. Exploring on Wikipedia I found out about the following two papers that give reasonable constants for those LCGs:

https://arxiv.org/abs/2001.05304

https://www.ams.org/journals/mcom/1999-68-225/S0025-5718-99-00996-5/S0025-5718-99-00996-5.pdf

LCGs are essentially defined by the following relation:

X(n+1) = (a * X(n) + c) mod m, n >= 0.

There are 4 constants involved: m the modulus, a the multiplier, X(0) the starting value or seed and c the increment. The problem is that the LCG constants tables in the papers only give combinations of good m and a. There is no mention about the X(0) and c constants that you need to choose.

Any help in understanding how to choose X(0) and c or mabe me not understanding the papers is much appreciated.

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I have emailed Pierre Lecuyer as well to ask about this problem and he replied exactly with the following points:

  1. m should be a prime number and c=0. The choice of a is crucial and not easy. See the papers. X(0) can be any number from 1 to m-1.
  2. However, LCGs like that are no longer recommended; they are not suitable for modern computers because they have too much structure and their period is too small.

He also gave me 3 references from his papers/books where I may find what I need, again exactly as he replied:

See chapter 3 of this draft book:

https://www-labs.iro.umontreal.ca/~lecuyer/ift6561/book.pdf

Also here:

https://informs-sim.org/wsc15papers/003.pdf

https://www-labs.iro.umontreal.ca/~lecuyer/myftp/papers/handstat2.pdf

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