1. The man who knew 28% of mathematics. John Von Neumann (1903-1957) http://www.scidiv.bcc.ctc.edu/Math/vonNeumann.html Hans Bethe: Academic seminars (10 levels) who can understand Level 1: my mother Level 2: my wife (This is sexism, my apology.) Level 7: myself Level 8: John and the Speaker Level 9: John, (the Speaker didn't) Level 10: not even Johnny

2. Random Numbers Generators “Any one who considers arithmetical method of producing Random numbers is, of course, in a state of sin…..” “… there is no such thing as a random number – there are only methods to produce random numbers, and an arithmetical procedure is of course not such a method…” “..... a problem we suspect of being solvable by random methods may be solvable by some rigorously defined sequence….” -- John Von Neumann --

3. Two dice Are they really random? But, there is no die in any computer. How to use them to generate random numbers? Roll one die: 4, 3, 2, 2, 6, 3, 5, 4, 6,... Roll two dice: 4, 8, 5, 9, 10, 2, 8,...

4. Image from http://www.ags.uci.edu/~bcarver/wgallery.html Ludwig Wittgenstein (1889-1951) “Turing Machines are human that compute.” “In logic nothing is accidental”

5. Using Computers A new random number will be generated based on some old numbers. Pseudo-Random numbers. The 1 st one is based on a seed. X0X0 X1X1 X3X3 X2X2 X i+1 XiXi X i-1 Criterion 1. How long is the period? 2. Is that sequence sufficiently random ?

6. Shift digits Method X i+1 =(X i  X i / 100 ) mod 100,000 X mod m = the remainder of X divided by m e.g. 9 mod 5 = 4, 5 mod 2 = 1, 9 mod 3 = 0. X0X0 X1X1 X3X3 X2X2 X i+1 XiXi X i-1 Suppose we want to have a sequence of random numbers between 0 and 99999 12345  12345 = 15239902523990  23990 = 575520100 12345 55201  55201 = 3047150401

7. Linear Congruential Method Let a, c, and m be integer. X i+1 =(aX i + c) mod m a : multiplier c: increment m: modulus X mod m = the remainder of X divided by m e.g. 9 mod 5 = 4, 5 mod 2 = 1, 9 mod 3 = 0.

8. Theorem: The linear congruential sequence has a period of length m iff 1. c is relatively prime to m; 2. (a - 1) is a multiple of every prime p dividing m; 3. (a - 1) is a multiple of 4, if m is a multiple of 4. The first proof was due to M. Greenberger in 1961 for m = 2 n ; The general case was proven by Hull and Dobell in 1962. X i+1 =(aX i + c) mod m