Re: Towards a Formula for Primes

On May 12, 1:40?pm, "charlesweh...@xxxxxxxxxxx"
<charlesweh...@xxxxxxxxxxx> wrote:
BEWARE, ALL WHO ENTER HERE, THE NOTORIOUS MINDLESS BICKERING OF
SCI.MATH.

I have at this point to correct one of my earlier statements.

Looks like you'll have to correct some later ones as well.

I do not
need bickering to help me. It helps nobody. I am perfectly capable of
correcting myself. I also accept the non-malicious counterexamples
from proper mathematicians. The mathematics decides.

I did a few preliminary studies of primes, in relation to binary
logic.

Something that is fairly obvious cropped up. If you XOR a number with
a different number, you can never get zero - the logical "false".

However, if you often get, for example, the number 1, you can subtract
1 from it to obtain the number zero.

I am not saying that this story that follows is exactly true, but I
know that George Boole was a school teacher. He may have said to one
pupil "What is 3 and 5", and obtained the answer "eight". He may have
said "CORRECT". He may have asked another pupil "multiply 3 and 5".
That other pupil may have said "eight", and been told "WRONG". He will
have replied "BUT YOU JUST SAID IT WAS CORRECT".

Perhaps triggered by such an incident, Boole seems to have pondered
the mathematical significance of "AND".

"Three AND five", in arithmetic, is the ADDITION. It represents the
verb to "ADD".

"Multiply three AND five" is a logical CONNECTIVE. The verb is not to
ADD, but to MULTIPLY.

Using 1 as a number for "true", and 0 as "false", he came up with

! * 1 is 1
1* 0 is 0
0 * 1 is 0
0 * 0 is 0

This is a TRUTH TABLE, but is also an ARITHMETIC TABLE.
Boole used the multiplication sign for logic "AND", and that continues
to this day.

With "AND" the first and second argument must be non-zero to get a non-
zero result.

For the OR table, we use addition

1 + 1 is 2
1 + 0 is 1
0 + 1 is 1
0 + 0 is 0

This is an arithmetic table. However, as ANY non-zero is taken as
"true", in Boolean arithmetic we have

1 + 1 is 1
1 + 0 is 1
0 + 1 is 1
0 + 0 is 0

One OR the other argument being "true" leads the result to be "true".

The + is used to represent "OR".

The 2 is changed into a 1 just to stop people being too preoccupied
with it as a number, rather than a "truth value".

So ARITHMETIC CAN DECIDE. I had previously believed the opposite.

So, by bringing XOR into the mathematics I have not added the power to
deliver a Boolean answer. Boole took his cue from arithmetic, and the
"decision-making" power came from there. Whether this is the original
Hilbert "decidability" is not yet clear to me.

The only thing that XOR seems to import into arithmetic is a (possibly
new to arithmetic) form of pseudo-randomness.

I wrote three tiny programs. The first displays the first 25 primes as
binary, followed by the 75 non-primes between 1 and 100.
They are written in Qbasic. Perhaps "Mix" would have been better.

The idea behind that is simply to see if some pattern can be spotted
in the binary that is unique to primes.

The next program uses XOR to combine a number with its previous. You
can see a kind of pseudo-randomness already.

No, you can't. It's not pseudo-random. Count the number of 1's in each
result. The counts are a 2-adic sequence (you forgot to do do 1 XOR
0):

1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,8,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,9,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,8,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,10...


The third uses XOR to combine a number with its double. Again, there
is pseudo-randomness.

The purpose is simply to become familiar with the behaviour of XOR.

Looks like you failed the test.


I did not invent axiomatic analysis.

I can tell.

That comes from the ancient
Greeks. However, before tackling the problem of primes, one has to be
sure the mathematical toolkit is complete. I am not convinced that it
is.

Charles Douglas Wehner


.

Relevant Pages

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