1
vote
2answers
43 views

How to create a random number generator that is completely random.

I'm trying to write an autonomous music composition program that makes use of many random number generators in order to make seemingly random melodies but I have no clue how to formulate a rigorous ...
12
votes
1answer
287 views

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
3
votes
1answer
122 views

Probability sequential terms of a linear congruential generator are coprime

I was following section 3.1.2 of the Structure and Interpretation of Computer Programming where they calculate $\pi$ using Cesàro's theorem that the probability two randomly chosen numbers are coprime ...
3
votes
1answer
282 views

Number of samples to predict the next number in a pseudorandom number generator

Let: $$R_{n+1} = (mR_n + b) \bmod{a} $$ Assume we know the values of $R_1, R_2, \ldots, R_L $. What is the minimum value of $L$ (if it exists) such that we can determine $R_0, m, b$ and $a$?
0
votes
1answer
87 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
11
votes
2answers
647 views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
3
votes
1answer
156 views

Randomness in prime numbers

I'm very interested in possible randomness in prime numbers distribution. There are many methods for "decomposition" regularity and randomness in primes (e.g. subtraction of some asymptotics , ...
1
vote
0answers
79 views

The mathematics behind Sobol sequences

I am using Sobol sequence as random number generator in a computer program. Beyond just making the program work, I would like to learn the mathematics behind the Sobol sequence (and other ...
0
votes
2answers
257 views

How to check that a sequence of numbers is random? [duplicate]

I have a sequence of numbers like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65... IS there any formula to check whether the sequence is random or not ? In my case odd numbers are ...
4
votes
2answers
508 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
0
votes
2answers
53 views

Lower bound on a function of probability distribution

Let $$ \gamma = \frac{1}{\sum_{y}f(y)W(y)}, $$ where $$ f(y) = 1 + e^{-|y|} $$ and $W(y)$ is a probability distribution (unknown) with $y \in \mathcal{Y}$ arbitrary but discrete, and $x \in ...
1
vote
1answer
81 views

Finding a subsequence (of a very long sequence) which does not sum to an even number

[Edited. I've revised to problem to focus on the special case of the integers modulo 2.] You are given a function f from binary strings x ∈ {0,1}n to the integers, or (without loss of ...
0
votes
1answer
179 views

Is the average of many “random” numbers useful information?

Ok, so I found this site: http://tweetcracker.com/. Essentially, people just tweet 10 digit numbers in hopes it is the correct number (like lottery, except free). I heard that if you took all the ...