Re: Towards a Formula for Primes

Continuing the story of the search for a general formula for primes -
even with the malicious "background music" that sci.math is notorious
for:

I generalised the concept of a system of mathematics that moves a
function across from the world of logic, and called such a system
"transarithmetic".

Here is such a formula, which I have used very often, such as when
writing a search engine or text comparator:

a = b AND 223

I have no doubt that very, very many people have done this also. I am
not suggesting therefore that "transarithmetic" has no prior Art. I am
suggesting, however, that it has not been subjected to a rigorous
axiomatic study.

The function I showed takes as its ARGUMENT (b) an ASCII number, and
uses the MASK 223 to reset the 32s bit, therefore delivering the
RESULT (a) that is upper-case ASCII. Yes - I know - there are
punctiuation marks &c.

By the discovery of pseudorandomness in primes, which arithmetic may
be unable to deal with, I recommended that the XOR function be moved
across into that arithmetic to create a "transarithmetic". Then, I
thought about this deeper.

The formula that is sought is of the form (NO, bickerers, this is NOT
the finished formula):

prime = (n + a - b / c * d) xor (n to the power e)

What this means is as follows:
(1)You are looking for "prime", the RESULT.
(2)You are using n, the ARGUMENT
(3)You have numbers such as a, b, c, d, e at your disposal - positive
whole number.
Those numbers cannot be an undiscovered prime, or you must search
for a prime to begin to search for a prime.
(4)Your functions are add (+), subtract (-), divide (/), multiply (*)
and XOR.
The "mirror-image twin" of XOR, the XNOR, can be substituted for
the logic function.
(5)Other functions, like the square, cube and (n to the power e) are
permitted, because they simply re-use the multiplication.

In the book in which he introduced the algebra, al Kwarizmi said:
"I have often considered what a person requires when doing a
calculation, and have decided it is a NUMBER".

Right, a number. But what KIND of number?

simplifying the above, we have
prime = f(n)

We feed IN a number n, and out comes the number "prime". It is a
BOOLEAN reply that we want. We want to know simply "IS IT PRIME"? More
would be less.

So, by importing a logic function into the arithmetic, I have not just
introduced tools that can handle the pseudo-randomness. I have
introduced the axiom of DECISION. This is the Babbage "inference",
from his "inference engine". It is the Boolean "truth", where (in
positive logic) zero means "false" but a number means "true".

We know that arithmetic cannot decide.

We know this because arithmetic existed for thousands of years before
George Boole.

Boole's contribution was to let zero represent "false", and at its
simplest, the number 1 to represent "true". This is the axiom of
decision. Had it already existed, Boole would have contributed
nothing.

So the reason why there is no general formula for primes is that the
axioms of arithmetic are not up to the task. If a formula is purely
arithmetical, it is guaranteed to be an approximation.

However, we have all the axioms we need in an "xor transarithmetic".
There is just enough logic in a set like +,-,/,*,xor to cover all the
parameters of primes.

We have the axioms. We have the toolkit. We do not (as yet) have the
formula.

Charles Douglas Wehner

.

Relevant Pages

  • Re: Towards a Formula for Primes
    ... I generalised the concept of a system of mathematics that moves a ... By the discovery of pseudorandomness in primes, ... I recommended that the XOR function be moved ... we have all the axioms we need in an "xor transarithmetic". ...
    (sci.math)
  • Re: The set of All sets
    ... zuhair wrote: ... >> MoeBlee ... ZFC has what are called axioms, ... T xor T => F ...
    (sci.math)