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### Parameters for LCG (linear congruential generators)

```y_n = a * y_{n-1} + b (mod p)     n > 0
```

Hint: A rule of thumb suggests not to use more than sqrt(p) random numbers from an LCG.

Notice that moduli larger than 2^32 require a computer with `sizeof(long)>32`.

Generators recommended by Park and Miller (1988), "Random number generators: good ones are hard to find", Comm. ACM 31, pp. 1192-1201 (Minimal standard).

 modul p multiplicator a ----------- -------------------------- 2^31 - 1 = 2147483647 16807 (b = 0)

Generators recommended by Fishman (1990), "Multiplicative congruential random number generators with modulus 2^beta: An exhaustive analysis for beta=32 and a partial analysis for beta=48", Math. Comp. 54, pp. 331-344.

 modul p multiplicator a ----------- -------------------------- 2^31 - 1 = 2147483647 950706376 (b = 0)

Generators recommended by L'Ecuyer (1999), "Tables of linear congruential generators of different sizes and good lattice structure", Math.Comp. 68, pp. 249-260. (constant b = 0.)

Generators with short periods can be used for generating quasi-random numbers (Quasi-Monte Carlo methods). In this case the whole period should be used.

(These figures are listed without warranty. Please see also the original paper.)

 modul p multiplicator a ----------- -------------------------- 2^8 - 5 = 251 33 55 2^9 - 3 = 509 25 110 273 349 2^10 - 3 = 1021 65 331 2^11 - 9 = 2039 995 328 393 2^12 - 3 = 4093 209 235 219 3551 2^13 - 1 = 8191 884 1716 2685 2^14 - 3 = 16381 572 3007 665 12957 2^15 - 19 = 32749 219 1944 9515 22661 2^16 - 15 = 65521 17364 33285 2469 2^17 - 1 = 131071 43165 29223 29803 2^18 - 5 = 262139 92717 21876 2^19 - 1 = 524287 283741 37698 155411 2^20 - 3 = 1048573 380985 604211 100768 947805 22202 1026371 2^21 - 9 = 2097143 360889 1043187 1939807 2^22 - 3 = 4194301 914334 2788150 1731287 2463014 2^23 - 15 = 8388593 653276 3219358 1706325 6682268 422527 7966066 2^24 - 3 = 16777213 6423135 7050296 4408741 12368472 931724 15845489 2^25 - 39 = 33554393 25907312 12836191 28133808 25612572 31693768 2^26 - 5 = 67108859 26590841 19552116 66117721 2^27 - 39 = 134217689 45576512 63826429 3162696 2^28 - 57 = 268435399 246049789 140853223 29908911 104122896 2^29 - 3 = 536870909 520332806 530877178 2^30 - 35 = 1073741789 771645345 295397169 921746065 2^31 - 1 = 2147483647 1583458089 784588716 2^32 - 5 = 4294967291 1588635695 1223106847 279470273 2^33 - 9 = 8589934583 7425194315 2278442619 7312638624 2^34 - 41 = 17179869143 5295517759 473186378 2^35 - 31 = 34359738337 3124199165 22277574834 8094871968 2^36 - 5 = 68719476731 49865143810 45453986995 2^37 - 25 = 137438953447 76886758244 2996735870 85876534675 2^38 - 45 = 274877906899 17838542566 101262352583 24271817484 2^39 - 7 = 549755813881 61992693052 486583348513 541240737696 2^40 - 87 = 1099511627689 1038914804222 88718554611 937333352873 2^41 - 21 = 2199023255531 140245111714 416480024109 1319743354064 2^42 - 11 = 4398046511093 2214813540776 2928603677866 92644101553 2^43 - 57 = 8796093022151 4928052325348 4204926164974 3663455557440 2^44 - 17 = 17592186044399 6307617245999 11394954323348 949305806524 2^45 - 55 = 35184372088777 25933916233908 18586042069168 20827157855185 2^46 - 21 = 70368744177643 63975993200055 15721062042478 31895852118078 2^47 - 115 = 140737488355213 72624924005429 47912952719020 106090059835221 2^48 - 59 = 281474976710597 49235258628958 51699608632694 59279420901007 2^49 - 81 = 562949953421231 265609885904224 480567615612976 305898857643681 2^50 - 27 = 1125899906842597 1087141320185010 157252724901243 791038363307311 2^51 - 129 = 2251799813685119 349044191547257 277678575478219 486848186921772 2^52 - 47 = 4503599627370449 4359287924442956 3622689089018661 711667642880185 2^53 - 111 = 9007199254740881 2082839274626558 4179081713689027 5667072534355537 2^54 - 33 = 18014398509481951 9131148267933071 3819217137918427 11676603717543485 2^55 - 55 = 36028797018963913 33266544676670489 19708881949174686 32075972421209701 2^56 - 5 = 72057594037927931 4595551687825993 26093644409268278 4595551687828611 2^57 - 13 = 144115188075855859 75953708294752990 95424006161758065 133686472073660397 2^58 - 27 = 288230376151711717 101565695086122187 163847936876980536 206638310974457555 2^59 - 55 = 576460752303423433 346764851511064641 124795884580648576 573223409952553925 2^60 - 93 = 1152921504606846883 561860773102413563 439138238526007932 734022639675925522 2^61 - 1 = 2305843009213693951 1351750484049952003 1070922063159934167 1267205010812451270 2^62 - 57 = 4611686018427387847 2774243619903564593 431334713195186118 2192641879660214934 2^63 - 25 = 9223372036854775783 4645906587823291368 2551091334535185398 4373305567859904186 2^64 - 59 = 18446744073709551557 13891176665706064842 2227057010910366687 18263440312458789471