Re: About random, primes and statistics

On Aug 20, 8:31 pm, JSH <jst...@xxxxxxxxx> wrote:
Knowing that statistics is an area of mathematics
that physics
students get rather familiar with I thought it'd be
of interest to
explain a simple area where mathematicians
routinely lie--I think in
order to keep research grants.

First you need to learn a bit of number theory, as
7 mod 3 = 1, where
1 is the residue modulo 3, so the bit of math is
that shown x mod y,
you take x-ky where k is the largest positive
integer that will fit,
and use what's left over, which is the residue.

I picked 3 because there is a fascinatingly boring
thing about numbers
modulo 3--perfect regularity:

Starting at 1 and counting up you have

1, 2, 3 followed by 4, 5, 6 followed by 7, 8, 9 on
out to infinity

which modulo 3 gives

1, 2, 0 followed by 1, 2, 0 followed by 1, 2, 0
repeated on out to
infinity.

(It's like a perfect waltz to infinity!!!)

That is an absolute and I'd say a trivial one at
that but it will
challenge everything you think you know about
mathematicians as decent
researchers as you have primes and you have
composites and composites
are products of primes--kind of like their
children!--so if primes
tended to pick a particular residue modulo 3, then
composites would
follow along!

But they don't. They split evenly between 0, 1 and
2.

Therefore, I strongly suggest to you, primes split
evenly between
having a residue of 1 and 2 modulo 3.

If that is true then the residue of a prime other
than 3 modulo 3 will
be random.

Here is what you get with the first 23 primes
greater than 3:

5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 =
1, 17 mod 3 = 2,
19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3
= 1, 37 mod 3 = 1,
41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3
= 2, 59 mod 3 = 2,
61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3
= 1, 79 mod 3 = 1,
83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1

So the sequence is

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1,
2, 1, 1, 2, 2, 1

and I suggest to you it is random, like flipping a
coin, but better as
it is absolutely random.



A sequence is random if there is
no way to predict the next element with a probability
greater
than chance.

Prediction rule.

If the current element is a 1 the next element
ment will be 2
If the current element is a 2 the next element
ment will be 1

This succeeeds with a probability greater than
chance.
The sequence is not random.

(Note that it is enough to give a prediction rule
that works.
It is not necessary to explain why it works)

- William Hughes




Yes, however assuming JSH meant pseudorandomness or randomness as in the digits of a number like pi() or any other normal number, he still has an interesting idea.

Unfortunately JSH has (as usual) only ideas and zero results.

Therefore i redirected him to "The One".

john
.

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