All Prime Numbers

**by Charles Dean Pruitt(with trigonometric prompting by Katya A.)ABSTRACT:The
following is a method to produce equations of various complexitieswith real coeffecients
whose solutions generate all prime numbers in an interval.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.Equation A. y = (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 by elimination 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 49 (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 (figure 6).

That curve is stated as:

y = [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 by elimination 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.)

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.

This not a true equation whose solutions yield primes.

To do this it is necessary to choose periodic equations

which are not as tidy as a Sine curve.

This is done by using a similar curve (in this case a Cosine curve)

and translating the axes. According to the parameters set out above,

a necessary condition is that the curve intersects the x-axis in a periodic fashion

and at the points where it intersects exhibits a desired characteristic.

In this case relative primality.

Consider the case of relative primality with respect to two.

This is straightforward since it is possible to construct a curve that intersects

the x-axis at points relatively prime wih respect to two (i.e. 1, 3, 5, 7, etc.).

Below is a Cosine curve with the axes translated such that the troughs of the

Cosine curve intersect the x-axis at integers relatively prime to two.

The equation of this curve is y = 1+Cos(Pi(x))

Likewise it is possible to choose similar Cosine curves for other integers.

However to do this for three for example, it is necessary to use multiple curves since a

single Cosine curve will not touch the x-axis at every integer relatively prime to three.

The first curve is y = 1+Cos(Pi(2x+5)/3)

and are all relatively prime to 3.

The second curve is y = 1+Cos(Pi(2x+7)/3)

and are all relatively prime to 3.

When y = 1+Cos(Pi(2x+5)/3) and y = 1+Cos(Pi(2x+7)/3)

are multiplied together the equation is

y = 1 + Cos(Pi2x+5)/3) + Cos(Pi(2x+7)/3) + Cos(Pi(2x+5)/3)*Cos(Pi(2x+7)/3).

y = 1 + Cos(Pi(2x+5)/3) + Cos(Pi(2x+7)/3) + Cos(Pi(2x+5)/3)*Cos(Pi(2x+7)/3)

and are all relatively prime to 3.

To get an equation which when solved for y = 0 yields values of "x" which are

relatively prime to two and three it is necessary to add Equation G and Equation J.

If y = 1 + Cos(Pi(2x+5)/3) + Cos(Pi(2x+7)/3) + Cos(Pi(2x+5)/3)*Cos(Pi(2x+7)/3)

is added to y = 1+Cos(Pi(x) the resulting equation is

y = 2+ Cos(Pi(2x+5)/3) + Cos(Pi(2x+7)/3) + Cos(Pi(x)) + Cos(Pi(2x+5)/3)*Cos(Pi(2x+7)/3).

When solved for y = 0, this equation gives values relatively prime to infinity and

gives prime values for "x" when y = 0 when x<5².

As is readily apparent, this equation is unwieldy and

not amenable for searching for primes.

The graph of this equation is:

y = 2+ Cos(Pi(2x+5)/3) + Cos(Pi(2x+7)/3) + Cos(Pi(x)) + Cos(Pi(2x+5)/3)*Cos(Pi(2x+7)/3)

respect to "y" are relatively prime to five are derived from the following equations:

y = 1+Cos(Pi(2x+7)/5)

y = 1 + Cos[Pi(7 + 2x)/5] + Cos[Pi(9 + 2x)/5] + Cos[Pi(7 + 2x/5] * Cos[Pi(9 + 2x)/5] +

Cos[Pi(11 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(13 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5)] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5]

The graph for this is:

y = 1 + Cos[Pi(7 + 2x)/5] + Cos[Pi(9 + 2x)/5] + Cos[Pi(7 + 2x/5] * Cos[Pi(9 + 2x)/5] +

Cos[Pi(11 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(13 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5)] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5]

yields an infinite number of values of "x" which are relatively prime with respect to

2, 3 and 5. For values of "x" less than 7², "x" is prime.

This equation is:

y = 3 + Cos[Pi(2x+5)/3] + Cos[Pi(2x+7)/3] + Cos[Pi(x)] + Cos[Pi(2x+5)/3]*Cos[Pi(2x+7)/3]

+ Cos[Pi(7 + 2x)/5] + Cos[Pi(9 + 2x)/5] + Cos[Pi(7 + 2x/5] * Cos[Pi(9 + 2x)/5] +

Cos[Pi(11 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(13 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5)] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5]

The graph for this is:

y = 3 + Cos[Pi(2x+5)/3] + Cos[Pi(2x+7)/3] + Cos[Pi(x)] + Cos[Pi(2x+5)/3]*Cos[Pi(2x+7)/3]

+ Cos[Pi(7 + 2x)/5] + Cos[Pi(9 + 2x)/5] + Cos[Pi(7 + 2x/5] * Cos[Pi(9 + 2x)/5] +

Cos[Pi(11 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] +

Cos[Pi(13 + 2x)/5] + Cos[Pi(7 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5] +

Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5)] +

Cos[Pi(7 + 2x)/5] * Cos[Pi(9 + 2x)/5] * Cos[Pi(11 + 2x)/5] * Cos[Pi(13 + 2x)/5]

it is obvious that it has a period of thirty (2 x 3 x 5).

Similarly the next curve would have a periodicity of 210 (2 x 3 x 5 x7).

Are these true formulae for generating primes?

Yes.

Solve for zero and they generate integer values of

that are prime less than a certain value.

For example: if the graphs are summed for the family of curves

up to and including 1 + Cos[Pi(A + 2x)/P(n)] then

all primes less than P(n+1) are the solutions of the equation.

Are they useful formulae?

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.

Of other utility value is the possibility to demonstrate via

substitution in these equations a number's primality or compositeness.

Again there is the computational intensiveness.

By way of example, Equation F can test

all numbers less than

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 computational intensiveness can be reduced

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

is an open question.

Comments: mammoth@htc.net

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