prime definition
The article below is about a new prime
definition. It is the first submitted article, from an author different than
SATOCONOR.COM
S. Chambers ‘A New Algorithmic Definition Of Prime
Number’ 5.1. (2006)
Full paper
SATOCONOR.COM Journal of Mathematics
A New Algorithmic Definition Of Prime Number
By Stephen Chambers
For comments: sbccc@prodigy.net
Version 2.0 March 1, 2006
Abstract:
In A New Definition of
Prime Number I propose a new contextual definition of prime number,
in which (Odd) primes are identified in relation to their flanking Evens. An
elementary arithmetic formula is applied to successive Even numbers; the
difference between consecutive results signifies the factorial make-up of
intervening Odds. Prime numbers are identified exclusively and uniquely by 0.
Composite numbers are identified by factorial type, according to a cerrtain
numerical scheme.
I argue that the traditional
definition of prime number (crudely, a number divisible without
remainder only by itself and 1) is theoretically disreputable
and computationally vacuous. It developed in tandem with the Sieve of
Eratosthenes, in which composite numbers are first identified positively, and
non-composites afterward identified negatively, by default. What is
theoretically disreputable about the definition is the dependence of prime
'building blocks' on composite superstructure, rather than conversely; and what
is computationally vacuous is reciprocal division by self and 1.
In contrast, the contextual
identification of prime number naturally and correctly precedes that
of composites, and coherently supports the concept of primes as 'building
blocks'. It provides a positive identification of prime number,
independent of issues of "exact" (i.e., remainder-less) divisibility.
It is simple, transparent and elegant; it is computationally rich. And it
supports Eratosthenes' Sieve entire by superior method, opening up new,
unexpected avenues to investigation.
Moreover, it obviates the
concept of primes as negative class of non-composites. W. V. O. Quine (Word
and Object) has written: "...even the mathematician, realist ex
officio, is always glad to find that some particular mathematical results
that had been thought to depend on functions or classes of numbers ... can be
proved anew without appealing to objects other than numbers." The
mathematician can be happier still when the class he mitigates is both negative
and foundational.
1.
Introduction
floor(x) yields the
greatest integer less than or equal to x.
E1
is an Even number, 10 or greater.
i =
3, 5… sqrt(E1) [ ‘sqrt(E1)’ for the square root of E1.
]
When we apply the elementary
algorithm
to
successive Even number (beginning with 10) we generate a series of values which
begins
a)
1
1 1 2 2 2 3 3 5 6 6
6 7 8 8 9 9 9 11 11
14 etc… ,
by
means of which we are able to ascertain the factorial make-up of successive
intervening Odds, by noting either:
1) change
of value and the corresponding difference between values (e.g., the
change from 3 to 5, and the difference 2 between
them) or
2) repetition
of value (e.g., the three repetitions of 6)
A change
of value indicates that the intervening Odd is some type of composite, while a repetition
of value indicates that the intervening Odd is prime. This may be recursively
expressed by subtracting from each new value in a) the value of its
immediate predecessor. The resulting series, beginning
b) 0 0 1
0 0 1 0 2 1 0 0 1 1
0 1 0 0 2 0 3 etc…
represents
a sort of factorial map, in which each ‘0’ represents a prime
position, each ‘1’ represents the position of either p3 or p*p
(where each separate ‘p’ represents a distinct prime), each ‘2’ represents the
position of either p2 or p2*p, etc.
To
better understand the underlying calculatory strategy, we will alter the
algorithm in the following way:
First,
remove the Summation (so as to display all individual values)
Next,
choose any particular Odd i (here, arbitrarily 3)
Finally,
apply this altered formula to successive Evens (beginning in this example with
2 and ending arbitrarily with 32).
E =
2, 4 … 32
This
gives a picture of the behavior of a particular Odd i (here 3) in
relation to successive Evens.
The grey-shaded column, to
left, is the index column. The value of E at any particular index is 2*index+2.
When
Odd i = 3 (as in this particular example) values are repeated thrice
(three 1’s, three 2’s, three 3’s, etc.). And in general, for any Odd i =
x, values are repeated x times. This is easily verified.
The
1st ‘0’ occurs, for any particular i, at index 
. In this
particular example, with i = 3, it occurs at index 1; when i
= 5, it occurs at index 2, etc.
A change
in value between two consecutive indices indicates that the Odd number between
them is divisible by i. For example, the change in value between index
6 and index 7 shows that the number between 14 (2*6+2) and 16 (2*7+2) is
divisible by the value of i (here 3). This obviously holds for all i.
Now, the Summation in the
original (unaltered) algorithm provides a tally of all such increases
due to changes in any i, and concomitantly a factorial tally for the
intervening Odd. When there is no increase, the intervening Odd lacks
divisors, and is prime. In this way we verify the procedure entire.
By similar reasoning
with
i added to E1 in the numerator is equally effective.
A more elaborate version of
the elementary algorithm, with focus on primes , is:
exp’
(short for ‘exponent’) is an integer > 0
E0
= E1 - 2
E1-1
is prime if and only if:
This
is either True or False, and identifies (algorithmically defines)
prime numbers exclusively. But it can, of course, be written
alternatively as:
and
this version lends itself to the identification of composites as well. If, for
example, E1-1 is the product of n distinct primes, then:
Thus,
in the particular case where E1-1 is prime, n=1, and 2n-1-1
= 0.
Or again, where E1-1
is primen (a prime multiplied by itself n times), we
find that:
Similar formulae can be developed
for all composites, but (depending on the formulation) the results are not
always distinct. For example, when the number of distinct primes n is
equal to 2, both 2n-1-1 and floor(n/2) yield a result of 1. This
particular formulation, therefore, does not differentiate the product of two
distinct primes from the square of a single prime.
By similar method, we can
exclusively identify (i.e., algorithmically define) Prime Pairs.
E2
is an Even number, 10 or greater
E0
= E2-4
i =
3, 5 … sqrt(E2)
E0+1
and E2-1 are Prime Pair if and only if:
Here is another useful
variant:
E1-1
is prime if and only if:
with
(i +1) subtracted from the Even in the numerator. Note that numerators are now
uniformly Even, and we can work directly with the Natural Numbers.
N1
is a natural number >5
N0
= N1-1
i =
3, 5…sqrt(2*N1-1)
2*N1-1
is prime if and only if:
I
began this web-page by applying a certain basic algorithm to consecutive Even numbers.
Later I applied a variation of it to the halves of Evens (i.e., consecutive
Natural Numbers). And now I apply a similar version to consecutive halves
of Natural Numbers.
For
incremental half-Units halfN beginning with 1.5 (viz., 1.5,
2, 2.5, 3, 3.5, etc.)
[ Note:
here the range variable n takes on integer values 0, 1, 2...etc. ]
If
is
True then 4 * halfN + 3 is prime.
Accordingly,
if
then
4*halfN+1 and 4*halfN+3 are Prime Pair.
The factorial mapping of
consecutive Odd numbers, as set out variously above, is simple, transparent and
elegant, yet with a curiously anachronic feel. The map is not a sieve. It is
possible, of course, to use these mapping methods to test particular, isolated
Odd numbers for their factorial make-up (such as primality), through comparison
of the algorithmic results of their flanking Evens. But there is nothing
sieve-like about the map itself; and this is where the greatest new potential
lies.
2.
Notation
The Factorial Map identifies
Odd numbers by structural characteristics—the structure of a prime number is
represented by ‘0’, the structure of the product of two primes by ‘1’, and so
on. I now propose a template Factorial Form for Natural Numbers, to aid
in the study of these structures.
Factorial
Form will also alleviate a problem with talk about ‘factors’—namely, that a
number may be said to have two or more factors (composites) or to have none
(primes), but never one factor and one only. This awkward numerical
discontinuity is usually sidestepped by talk instead of divisors – for
example, a prime number, in the traditional definition, is said to be divisible
(not ‘factorable’) by 1.
3.
Factorial Form
Every Natural Number can be
represented as the product of two component multipliers, Even and Odd,
as follows:
y, x and n are
integers, and
|
|
|
|
|
|
Factorial Form:
12y * (n1px1 * n2px2 * n3px3 * ... * nkpxk)
1)
12y represents the Even
component of a Natural Number (including 120 = 1). The prescript 1 is constant and
purely formal (it is computationally vacuous); its only purpose is to smooth
out theory (vide infra). It represents a numerical value in the
usual way (e.g., 125 = 32, 120 = 1).
2) The
parenthetically grouped Odd component builds from expressions of the type
npx
in
which the letter ‘p‘ is a graphic name (short for ‘prime’) which serves
merely to consolidate, or bind, prescript n with superscript x.
Per
convention, px represents an unspecified prime to power x.
But the proposed new prescript n represents the number of distinct
primes raised to the specific exponent x (thus the numbered
prescripts n1, n2…nk and exponents x1, x2…xk). The notation
3p2
for
example, represents a product of three distinct primes, each with exponent 2
(e.g., 132*892*412, or 372*32*232).
But it conveys no information about the actual value (or order) of constituent
primes (in contrast to the Even component).
‘p’
obviously refers only to Odd primes(s); Factorial Form segregates 2 and
its self-multiples from Odds. Conceptually, this segregation conforms nicely
with the contextual definition of (Odd) prime numbers in terms of their Even
flanks.
To represent a purely Even
number 12y by means of Factorial
Form, we computationally vacate the Odd component by writing it 1p0 (i.e. 1). Conversely,
to represent a purely Odd number ( n1px1 * n2px2 * n3px3 * ... * nkpxk )
we computationally vacate the Even component by writing it 120 (again 1). We write,
for example:
120 * 1p0 for 1
121 * 1p0 for 2
120 * 1p1 for 3
Note
that the Form for 3 is the general Form for any single (Odd)
prime; but the Forms for 1 and 2 are specific and unique.
Every natural number is thus
schematically minimally two-part. The number of factors in an expression
is the sum
1*y
+ ((n1*x1) + (n2*x2) + (n3*x3)
+ … (nk*xk)).
The
purpose of the prescript
Here
are two examples:
120*3p1*2p2*2p3 = (1*0) +
(3*1) + (2*2) + (2*3) = 13 [ e.g., 17*11*7*52*132*33*193
]
123*2p1*3p2 = (1*3) +
(2*1) + (3*2) = 11 [ e.g., 23*31*5*72*412*132
]
This new characterization of factor
differs somewhat from the old. A single prime, for example, has 1 factor, viz.,
120 * 1p1 = (1*0) + (1*1) = 1
as
does also the number 2, viz.,
121 * 1p0 = (1*1) + (1*0) = 1
The
number 1, however, has no factor.
120 * 1p0 = (1*0) + (1*0) = 0
In terms
of Factorial Form the number 2 has one factor, in common with Odd primes; it is
treated separately due to its special parity. The status of 1 remains open,
since computationally 120 is indistinguishable
from 1p0. And composites all follow the
conventional scheme - 15 has two factors (3*5), 45 has three (5*32),
etc.
Nowadays, at any rate, 2 is
conventionally most often counted as prime. Strictly according to the
traditional definition of prime number, there is no reason to exclude it, but neither
is there reason to exclude 1. The question cannot be resolved in terms of the
traditional definition of prime number, but outside help is needed (this is a
weakness of the traditional definition of prime number).
4.
Prime Pairs
The
mapping methods above provide, for the first time, a deterministic test for
Prime Pairs. This might be useful for resolving the question whether there are
infinitely many of them.
Remarks:
There
is one and only one Prime Triple, consisting of the three specific consecutive
(Odd) prime numbers 3, 5 and 7.
Two
distinct primes p1 and p2, where the absolute value of (p1
- p2) is equal to 2, are often called “Twin Prime”. The
nomenclature is unfortunate in that it does not easily adapt to two (Odd)
primes separated by (Even) Intervals greater than 2. The other familiar name -
‘Prime Pair’ - is only a little better (I have used it as a model for ‘Prime
Triple’), but it too lacks a regular linguistic development.
Since
it is unresolved whether there are infinitely many Prime Pairs, it is
concomitantly unresolved whether there are infinitely many pairs of primes
separated by (Even) Interval 4, by Interval 6; etc. These questions are all of
a piece; and the resolution of any particular one could serve as a template for
resolution of the others.
5.
The Goldbach Phenomenon
Goldbach’s
Conjecture holds that every single Even number is the sum of two primes. The
factorial mapping method identifies a single prime contextually, in terms of
its two flanking Evens.
Remarks:
No-one,
probably, believes Goldbach’s Conjecture to be false - all existing evidence,
after all, supports its verity. What intrigues us about the Conjecture - what
makes us uneasy - is not the phenomenon itself (for surely it seems entirely
plausible) but that it so adamantly resists proof.
Imagine
that we found an Even number which failed Goldbach’s Conjecture and discredited
it. That is, imagine that we found not proof but empirical evidence, got by experiment,
that vacates even the possibility of proof. Whereas proof positive of the
Conjecture could show us a method, or path, useful for application to related
problems, the empirical failure of it would leave us empty-handed. We could
even end up saying “Well, Goldbach’s Conjecture turned out not to be a
problem at all!”. For although we can discredit by experiment, we cannot
similarly credit. There never comes a time when we say “Well, this is enough
evidence. I now accept Goldbach’s Conjecture as true”.
Our
inability to prove Goldbach’s Conjecture is attributable to either 1) a failure
of our individual problem-solving skills, challenged within an intrinsically
complete and effective mathematical system or 2) a fundamental lack in the
mathematical system itself, which no proficiency of computational skill could
possibly overcome.
We
cannot know in advance which of these two possibilities is at fault.
Ordinarily, a mathematician or logician who attempts to understand the
Conjecture is wise, at least initially, to question his own problem-solving
skills. But Goldbach’s Conjecture has resisted proof for so long, and has been
attempted by so many brilliant minds, that the latter seems more likely. Some
fundamental, essential piece of information appears to be entirely missing from
our computational palette.
In
such a situation it is necessary, in my opinion, to turn decidedly away from
proof (for proof is entirely embedded in the very system wherein lies the
hypothetical fault), and look instead for pure phenomenon, keeping in mind that
phenomena can be demonstrated, but not proved. Let me elaborate
with an example:
The
phenomenon of linearity - geometric “straightness” - is initially learned via
demonstration, ostention, etc. The mathematical model of it, which comes only
afterward, teaches us nothing new about “straightness”, but rather about the
potential, versatility and adequacy of the mathematical modeling-system itself.
One would hardly realize this from the way we talk.
6.
Floor
& Mantissa functions
Values generated by the
Mapping Algorithm sans floor function exhibit two distinct parts, 1)
integer and 2) fractional (below, in decimal notation). The Mapping Algorithm
compares integer values, via the floor function—identical values for
(consecutive) E0 and E1 indicate that E1-1 is
prime.
We
now turn our attention to mantissa (fractional) values for consecutive
Even numbers E0 and E1 (below, using arbitrary exemplary
values 88 and 90).
mantissa(x)
= x – floor(x)
E0
= 88 E1 = E0
+ 2 (i.e., 90)
|
|
Consecutive
values differ by 1/i, that is:
For i factor of Ew:
Otherwise Ew
is of the form 12n*prime (including the vacant prime 1p0)
In
these particular examples, E1 = 90 exhibits factors 3, 5 and
9; and E0 = 88 is of the form 12n*prime
(viz., 23*11).
7.
Fundamental Equalities
First, note the relation
between mantissa and modular expressions:
And
next, the symmetry of numerators in the equations:
A:
is True
only for factors of E1-1.
B:
is True
only for non-factors of E1-1.
Note
that:
1)
Not all A are True (some A are False)
2)
If A is False for all i, then E1-1 is prime.
3)
Not all B are False (some B are True)
4)
If B is True for all i, then E1-1 is prime.
Similarly,
if the following are True for all i, then E1-1 is
prime; but if False for any i, then i factors E1-1.
1)
2)
Now compare the left and
right side of example 1) above, using new values for E0 and E1:
E0
= 86 E1 = E0 + 2 (i.e., 88)
For every factor i of
E1-1, E1 is negative (E0 is always
positive). We therefore need only check for negative values in E1 to
ascertain the primality of E1-1. That is:
If
the following is True for particular i, then i is a factor
of E1-1.
3)
Conversely,
if the following is True for particular i, then i is not
a factor of E1-1.
8.
The Traditional Definition of Prime Number
The traditional definition of
prime number, viz.
“a prime
number is divisible without remainder only by itself and
is
computationally vacuous. It describes a defining characteristic of prime
number, but does not tell us how to find one. It encourages a “sieve-like”
approach to the identification of prime number – that is, we first
describe a network of composites, and afterwards characterize primes as
non-composite. Yet we are taught (with good reason) that prime numbers are the
“building blocks” of Number Theory. Surely it is theoretically disreputable to
define a “building block” as a negative of the very superstructure which it is
said to support.
These
are weaknesses of the traditional definition. In contrast, there are compelling
positive reasons to urge a contextual identification of prime structure,
as set out in this site, as an outright Definition of Prime Number:
1)
The Factorial Map identifies not only prime structure but every
factorial structure, continuously and positively. In contrast, even
deterministic tests characterize primes in vacuo from composites.
2)
The contextual characterization of factorial structure is computationally
powerful, capable of supporting existing Theory. It provides a common “starting
point” for investigation and development, and an organizational nexus for
Factorial Theory.
-o0o-