# What are congruent linear pairs

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Linear congruence generators

- Form the starting point for most simulation algorithms
*Standard random number generators*, - A widely used method for generating standard pseudorandom numbers is as follows
*linear congruence method*,

We now solve the recursion equation (1), i.e., we show how the number recursively defined in (1) can be *directly* by the start value as well as through the parameters , or. expresses.

**proof****Notice**-
- It is clear that with the linear congruence generator defined in (1) at most different numbers can be generated.
- In the case of an unfavorable choice of parameters , , or. can be the length of the period be very small.

We now mention (sufficient and necessary) conditions for the parameters , , or. that in particular ensure that the maximum possible period is achieved.

**Theorem 3.2**

- 1.
- If , the in defined linear congruence generator exactly then for each starting value a sequence of numbers with the maximum possible period if the following conditions are met:
- (a)
- The parameters and are coprime.
- (a)
- For every prime number , the shares is a multiple of .
- (a)
- If a multiple of is, then is too a multiple of .

- 2.
- If , then applies for each exactly when
- (b)
- is a prime number,
- (b)
- for every prime number , the divides that number not through is divisible.

- 3.
- If and if there is one With there is exactly when is an odd number and if or applies.

- One
*proof*of Theorem 3.2, in which the results of number theory (including Fermat's Little Theorem) are used, one can for example- in section 2.7 of the book by B.D. Ripley (1987)
*Stochastic simulation*, J. Wiley & Sons, New York or - in Section 3.2 of D.E. Knuth (1997)
*The Art of Computer Programming*Find, Vol. II, Addison-Wesley, Reading MA.

- in section 2.7 of the book by B.D. Ripley (1987)

- In addition, we also refer to these two books for discussion
- One such quality characteristic is
- Further details can be found, for example, in the aforementioned book by Ripley (1987) or in the lecture script by H. Künsch at ftp://stat.ethz.ch/U/Kuensch/skript-sim.ps, where the the following figures are included.

**Next page:**Tests of quality properties

**Upwards:**Generation of pseudo random numbers

**Previous page:**Simple application examples; Monte Carlo Estimator & nbsp

**content**Ursa Pantle 2003-09-29

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