Primality or Compositeness

**by Charles Dean Pruitt(with trigonometric prompting by Katya A.)ABSTRACT:The
following is a method to produce formulas of various complexitieswith real coeffecients
whose itereation indicate whether a given integeris prime or composite.METHOD:From
theory and by inspection, composite numbers with similarfactors appear on the
number line in a periodic fashion.Conversely prime numbers appear on the number
line irregularly as "not composites".Therefore it is possible to find
a periodic function such as a sine curve or cycloid witha particular periodicity
which will cross the x-axis (on a two-dimensional Cartesiancoordinate system)
at each integer whose factors correspond to that periodicity.For purposes of
this argument the sine curve is used because it is the most familiar.For example:
the sine curve (Sin xPi/2) has an amplitude of one and passes throughthe points
on the x-axis (0,0), (2,0), (4,0), (6,0) to infinity, i.e. multiples of two.**Figure
1. (Sin xPi/2)

the points on the x-axis (0,0), (3,0), (6,0), (9,0) to infinity, i.e. multiples of three.

the points on the x-axis (0,0), (5,0), (10,0), (15,0) to infinity, i.e. multiples of five.

all points on the x-axis which have its respective prime (and periodicity) as a factor.

When Figure 1 is multiplied by Figure 2, Figure 4 results.

This is the graph of the curve (Sin xPi/2 * Sin xPi/3).

It is possible to see that the zero points where the curve crosses the x-axis

correspond to the integers which have 2 or 3 or both as a factor.

Where the curve does not intersect the x-axis at the integer values

are integers relatively prime to two or three.

This is the case up until 'x' equals 25 or five squared.

This is consistent with standard number theory concerning factorability.

Therefore for the equation

Sin xPi/2 * Sin xPi/3 = 0,

the points which satisfy the equation for 'x' are

(2,0), (3,0), (4,0), (6,0), (8,0), (9,0), (10,0), (12,0), (14,0), (15,0),

(16,0), (18,0), (20,0), (21,0), (22,0), (24,0).

By inspection (see Figure 4), it can be seen that there is no intersection with

the curve at the points on the x-axis

(5,0), (7,0), (11,0), (13,0), (17,0), (19,0), (23,0).

This is confirmed when these values for 'x' are substituted into

the equation and non-zero answers result.

In effect these are all the primes less than five squared except two and three.

By the above method is it possible to generate equations which when solved for

x =0 indicate that all integers left over are primes.

will cross the x-axis for all non-primes less than seven squared,

it is necessary to multiply (Sin xPi/2 * Sin xPi/3 * Sin xPi/5).

it is possible to recognize the prime points on the x-axis.

Therefore when solving for

Sin xPi/2 * Sin xPi/3 * Sin xPi/5 = 0,

the solutions for 'x' (when x<49) comprise

less than 7 squared as well as 2, 3, and 5.

Therefore all the remaining integers are prime.

With 2, 3, and 5 this constitutes all the primes less than 49.

It is possible to check this empirically by substitution.

For exceedingly large samples it is not useful to view a whole graph

because the fine details are not readily clear.

However it is possible to calculate for a large range

by multiplying all the consecutive sine curves together.

In the following example the first 27 sine curves (Sin xPi/2) through (Sin xPi/103)

were multiplied together to yield a graph for the resultant curve

of which 100 units are shown below.

That curve is stated as:

Sin xPi/2 * Sin xPi/3 * Sin xPi/5 * Sin xPi/7 * Sin xPi/11 * Sin xPi/13 * Sin xPi/17 *

Sin xPi/19 * Sin xPi/23 * Sin xPi/29 * Sin xPi/31 * Sin xPi/37 * Sin xPi/41 * Sin xPi/43 *

Sin xPi/47 * Sin xPi/53 * Sin xPi/59 * Sin xPi/61 * Sin xPi/67 * Sin xPi/71 * Sin xPi/73 *

Sin xPi/79 * Sin xPi/83 * Sin xPi/89 *Sin xPi/97 *Sin xPi/101 * Sin xPi/103

composite numbers up to 103 squared (10609) and

consequently all of the primes less than 10609.

is blown up (figure 7) to shown the fine detail and confirm

visually that indeed the resulting curve crosses the x-axis

at all the composite numbers and not at any of the primes.

However for all values of 'x' which satisfy the equation,

results can only be derived by solving for zero.

are prime (and constitute a prime pair). This is indeed the case.

The curve between 10004 and 10006 has an amplitude of less than one ten-billionth.

With further magnification (see Figure 8) details as fine one part in a trillion

are resolved and indicate 10005 is composite.

(Of course there are easier methods to determine

compositeness of numbers ending in 5.)

to construct an appropriate equation of periodic curves multiplied together

as above so that the final curve multiplied with all the initial curves

is Sin xPi/A where 'A' is the first prime greater

than the square root of the number being tested.

As a formula this reads:

Sin xPi/2*Sin xPi/3*Sin xPi/5*...*Sin xPi/A

Substituting the number to be tested for 'x' yield zero if the number

is a composite and a real non-zero value if the number is prime.

PROOF

Each periodic function as defined above intersects

the x-axis at points that correspond to all

the integers divisible by that function when y=0.

Sin xPi/2=0 represents all integers divisible by 2.

Sin xPi3=0 represents all integers divisible by 3.

Sin xPi/5=0 represents all integers divisible by 5.

When two functions are multiplied together,

they will show up all the zeroes because 0 times anything is 0 .

Sin xPi/2 * Sin xPi/3=0 represents all integers divisible by 2 and/or 3.

Sin xPi/3 * Sin xPi/5=0 represents all integers divisible by 3 and/or 5.

Sin xPi/5 * Sin xPi/7=0 represents all integers divisible by 5 and/or 7.

There is not duplication because 0 times 0 is 0

and hence there would be only one point indicated instead of two.

Likewise the same argument applies when many sine curves are multiplied together.

When all sine curves (each with periodicity corresponding to a prime and

all such curves up to and including that whose periodicity is

less than or equal to the square root of the largest number

in the sample space being tested) are multiplied together,

the resultant equation gives each composite number one time

when the equation is solved for zero.

Discarding these from the set of integers leaves the primes.

In practice this has become a formulaic Sieve of Erastathenes.

COMMENTARY

Are these true formulae for testing primality? Yes.

Are they useful formulae? In their current form they are trivial.

They are computationally intensive, but the idea of multiplying

periodic equations together and solving for zero has utility value.

Suppose periodic equations could be constructed that were inherently simple

and when multiplied together were likewise simple.

The restraint of computational intensity would be diminished

with a clever enough set of periodic equations.

If non-trigonometric periodic equations could be found,

it might produce a polynomial which could be solved for

and produce integer roots all of which are composite.

From this would come all the primes via solution of a polynomial

and subsequent cataloging those integers which don't solve the equation.

By way of example, Equation A can test

all numbers less than 10609 for primality or compositeness.

Substitute the number to be tested for 'x'.

If the answer is zero the number is composite.

If the answer is not zero the number is prime.

As mentioned before, the possibility to find simpler or

more clever periodic equations would reduce the computational intensity.

Another possibility is to choose periodic formulas where the

results can be simplified using trigonometric identities.

Whether the computation intensiveness can be reduced

low enough (i.e. linear instead of multiplicative or exponential)

is an open question.

Comments: mammoth@htc.net

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