Reddit mentions of Uniform Distribution of Sequences (Dover Books on Mathematics)

> Does it work only for few 'a' and 'c' ?

Yes, but it depends on the modulus too. For prime modulus, a lot of different combinations will work (i.e. give statistical properties as good as you can expect from an LCG). For power-of-2 modulus, most choices will not work (e.g. think about iterating 2x + c mod 32). In general, the problems arise when the modulus and the multiplier have common divisors, so a highly divisible modulus is already incompatible with most multipliers and increments. In particular, an even modulus rules out a full 50% of possible values.

Unfortunately, even determining the period of a given LCG requires a great deal of thought and a good chunk of elementary number theory: you can see e.g. Knuth, The Art of Computer Programming, vol. 2.

Interesting statistical properties are harder than that. It's possible to give conditions for n-dimensional equidistribution for given multipliers and the increments, but in practice it's easier to do a computational search for the good values. If you're interested in results of the form "these given families of multiplier-increment-modulus combinations will give bad results", I'd suggest looking into the works of Pierre L'Ecuyer.

I think you'll find reading the book of Kuipers-Niederreiter generally interesting. For measuring discrepancy LCGs, see this paper. The Mersenne Twister was created specifically in a way that makes even high-dimensional equidistribution easy to prove: you may read the Mastumoto-Nishimura paper to see how multiple-recursive matrix methods achieve that.