Prime numbers

Prime numbers,the Muses ,Apollon

Zarurile Afroditei

Aparent e un nou joc.un joc nou de societate,totusi nu dores sa intru in competitie cu   Antoine Court de Gébelin

Aleister Crowley

Marsilio Ficino.

Aparent e un nou joc...

un joc de perspicacitate,unde numerele prime nontriviale sunt liniile (corzile,ca in teoria corzilor ) ,numerele prime triviale membranele...Un joc unde chiar si legea lui Titius-Bode poate fi exprimata doar cu numere prime....un joc ....

Functia spectacolului,,,6 ,sase Muze...si Zeita Artemis

Heraclit ,lira si arcul

Lira lui Apollon si arcul lui Artemis.
Mistica lui Heraclit....si secta lui Pytagora.
Identitatea contrariilor.
Moartea initiatica.
Armonia
Muzica sacra si Templul (legile apolliniene pentru construire a Templului , a scenei, a corului,a imnului, a muzicii...etc..rtc. )

Tetraktys

Ciurul Teleman si geometria sacra

Numere perfecte,numere prime,numere prime triviale,adica produse de numere prime nontriviale.Aceste numere sunt suprafete.....dreptunghiuri frumoase....utilizate de artisti,plastici sau muzicieni....sau ingineri adevarati care stiu ca SIMETRIA si PROPORTIONALITATEA face o constructie durabila,fiabila si frumoasa.

Dreptunghiuri frumoase si paralepipede magnifice.

Numerele prime nontriviale sunt liniile,iar cele triviale spatiile ,cu doua sau trei dimensiuni.

Nicomachus was a Syrian mathematician writing about 150 A.D. His work forms one of the best links to what survived from his day about Greek theory of numbers and music [1,2]. I shall describe how the sequence of integers shown in Table 1, and attributed to Nicomachus, defines musical octaves, fifths, and fourths the only consonances recognized by the Greeks, and lies at the basis of ancient musical scales sometimes attributed to Pythagoras. A second Table inferred by Plato but brought to light by the ethnomusicologist, Ernest McClain [3,4,5], will be shown to be the basis of the Just scale, another ancient musical scale. This table, which I shall refer to as the McClain Table, will also provide a link to the modern theory of music. In his books and papers, McClain has made a strong case for music serving as the lingua franca of classical and sacred texts, providing plausible explanations to otherwise difficult to understand passages and providing metaphors to convey ideas and meaning. In this paper I will focus primarily on the mathematics and music at the basis of ancient musical scales.

.............

Sacred Mathematics represents a wholly contained and internally consistent philosophy -- a philosophy which describes physical reality, its cosmogony and every aspect of its science.  The Greek Philosopher, Plato, has said: “Geometry is knowledge of the eternally existent. Numbers are the highest degree of knowledge.  It is knowledge itself.”           

This profoundly philosophical interpretation or Mathematical Theory is echoed in many spiritual traditions.  Such references are not always direct, however, and in fact may have been intentionally obscure in order to prevent the “uninitiated” from gaining access to the power inherent in such knowledge.  The School of Pythagoras, for example, was known to limit the dissemination of the mathematical and philosophical understandings they achieved.  The so-called Mystery Schools of ancient Egypt, Sumeria, Greece, and Judaea were also very much into keeping their own counsel and limiting access by the uninitiated.  The biblical prohibition from eating of the Tree of Life or Tree of Knowledge of Good and Evil may represent another case of a little knowledge being a dangerous thing -- at least to someone's thinking.  

The same can be said of the modern world, where mathematics are taught in such a dismal and inadequate manner, that mathematical anxietyhas become a psychological condition semi-officially recognized by the public school systems, and in some respects encouraged as a means of convincing the majority to avoid like the plague the very idea of numbers and geometries.  Accordingly, the odds are very good that those who think of themselves as mathematically untalented (those who might quickly refer to A Non-Mathematical Digression, for example), may find they know far more mathematics than they might otherwise have expected.  Music, for example, is a primary illustration of the ability of individuals to unknowingly appreciate geometry and mathematics, even when they’re not serious left brain thinkers.  

The key in this regard [pardon the pun] is that Music is basically about ratios, frequencies (geometrical sine waves), and timing (a very mathematical kind of thingamagig).  There is also a strong geometrical connection, in that, if one takes the unique 3-4-5 right triangle (with sides of 3, 4, and 5 equal units -- the only triangular shape utilizing a combination of single digit whole numbers which result in one angle of the triangle equaling 90 degrees -- however, larger numbers such as 5, 12, and 13 also work), and strings a continuous fine wire to each of the three points of the triangle, it is then possible to tune one of the sides to a particular note, and have the other two sides be in a tuned harmony.  The three sides of the triangle form a series of tones that are equivalent to the first three strings of a tuned guitar.  (One might also note that geometrically, one only has to tune one side of the triangle, while the guitar requires all three strings to be individually tuned!)  

Other examples of geometrical knowledge not always appreciated include:  The well known Vesica Pisces (at least in one of its interpretations),The Great Pyramids (which are readily visualized and understood), and diamonds of various cuts (i.e. geometries). The latter are often fully understood and greatly appreciated by the mathematically disinclined (if not, in fact, a set of geometries quickly and accurately appraised from a financial view point by the truly discerning individual -- i.e. most of the females on the planet).  Initially, one may not be familiar with the distinction between a dodecahedron and an icosohedron (or a Rose cut), but once enlightened as to their definition and practical aspects, examples of the geometries suddenly begin showing up most everywhere.  

Sacred Mathematics is not exclusively about geometry, however.  There are, for example, Magic Squares , Fibonacci Numbers, numbers in general (e.g. Nines), Numerology, and a whole host of other strange relationships among the very real aspects of the universe.  

Pythagoras (of Pythagorean Theorem fame) taught that the motions of the planets, the Moon and Sun (as well as an invisible Anti-Earth on the other side of the Sun), created the Harmony of the Spheres, but which ordinary people could not hear because they were too accustomed to it.  [Either that, or an ancient Napster got caught in the act!]   

Pythagoras and his followers made no distinction between music, mathematics and magic. Magic and music, for example, were found to be based upon mathematical laws (albeit, sometimes very subtle ones).  Furthermore, everything in the universe followed these mathematical laws and the universe was created out of the geometrical relationships of the numbers, and thus constituted the true basis of reality.  Which says a lot.  

Pythagoras believed that each number was holy and had its own powers. One was the indivisible monad, creating everything out of itself.  Two was the pure duality, perfect balance between opposites. Three was the number of the gods, while four was the number of the material world (hence the four elements). And so on. This became the foundation of the Numerology that the Order of Hermes ultimately adopted.  The Tarotis also strongly reminiscent of Pythagorean thought in this regard.  

Plato -- in addition to being a geometry nut (the Platonic Solids are named after him; see also A Graphics Description) -- was also into the numbers.  So to speak.  After Socrates had been executed -- allegedly for the horrendous crime of accepting money from those he taught! (Imagine!) -- Plato left Athens and traveled to Egypt, Sicily and Italy.  At the latter pit stop, Plato learned of Pythagoras, and quickly came to appreciate the value of mathematics.  Based on ideas Plato gained from Pythagoras’ disciples, Plato decided:  

            “...that the reality which scientific thought is seeking must be expressible in mathematical terms, mathematics being the most precise and definite kind of thinking of which we are capable.”  <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html>  

Plato even came to believe strongly in the importance of the numbers 216 and 12,960,000.  There was no indication in Plato's writings as to whythese two numbers were important, but we can note that both reduce to Nine.  (See Numerology for an explanation of reducing numbers.) However, as one reader was quick to point out, 216 = 6 cubed), while 12,960,000 is 60 to the fourth power. This is all very Sumerian, in that 6 and 60 were part of their segesimal mathematics. What is probably not particularly Sumerian, however, is the curious fact that one could plausibly write: 216 = 6x6x6, or dropping the multiplier symbols (a common practice in mathematics and science), one obtains 666. This is, of course, the number of the beast in Revelations -- as well as the number of gold talents received in a year by King Solomon. Apparently,Revelations was not enamored with either Plato, King Solomon, or the Sumerians. (Alternatively, the "beast" in Revelations is man, and man being an animal... perhaps the most dangerous beast is simply man.)

As for 60606060... Perhaps it has something to do with an equilateral quad-angle in four dimensions with 60 degree angles are all corners.

Theon of Smyrna put together a handbook for philosophy students of Plato which showed how prime numbers, geometrical numbers such as squares, progressions, astronomy, and music were interrelated.  With respect to numbers Theon went Pythagorean, and discussed odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers.  Clearly, Theon’s days were numbered.  Link to: <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Theon_of_Smyrna.html> for more information.

More recently, Stan Tenan <www.meru.org> in his treatise, “The God of Abraham, A Mathematician’s View”, notes that, “Each and every culture has made accurate and effective models of the cyclic, self-propagating and self-referential nature of all life in terms appropriate to its needs and experience.”  In other words, the idols of sophisticated people are the cultural, social, and scientific Paradigms of their societies.  On this basis, Abraham “acts as a mathematician: he postulates a meaningful and functional definition of Unity.”  [emphasis added]  

Mathematically, any wave shape can be derived from an Infinite Series of simple sine waves.  Typically, an approximation is accomplished by using only the larger terms of the series, but even in the case of a “square wave”, a finite number of sine waves can provide a close mathematical approximation.  It follows by extrapolation, according to Tenan, that an infinitely loud, short, sharp pulse (an “Om” or “Big Bang”, which might be compared to a musical or mathematical singularity) would produce the harmonic spectrum of all tones.  The latter could be considered to be equivalent to All-That-Is.  

To add a bit of scientific credibility to all of this, we might note also that quantum physics -- the state-of-the-art version -- is now fully supportive of Sacred Mathematics.  Lothar Schäfer, for example, in discussing the concept of Causality, has noted that, “Epistemic principles are transcendental because they are neither derived by a process of reasoning, nor my operations performed on physical reality.  They are, simply, principles of the human mind.  Thus, identity, object permanence, causality, external reality -- all the requisites for a resonable and enlightened life, albeit uncertain to experience and reason -- are valid because they are transcendental principles provided by the human mind.  By producing these principles, it is as though the mind remembered a higher order than can be found in the laws of logic or the visible patterns of physical reality.  Thus, it is a valid question whether evidence can be found from physical science of this transcendent part of physical reality, where such a higher order might have its roots.” [1]  

More specifically, quantum waves are a third type of wave (in addition to matter waves and electromagnetic waves -- sound and light and so forth).  These waves are not only non-material -- i.e., needing no material medium in which to propagate -- but they are empty.  “Lightwaves can travel in empty space, but they carry energy.  Quantum waves also exist in empty space, but carry no energy or any other mechanical quantity.” [1] 

In other words, quantum waves are simply numbers, numerical relationships.  “Because they are empty, evidence of their existence is circumstantial; we must think that the universe is a network of quantum waves because the observable order appears as a manifestation of their interference.”  Also, “The reality of quantum waves is inferred from the expression of their interference in the observable patterns of reality.” [1]  

Forms, patterns, geometries, ratios...  

What an electron does is based on probabilities, which are in turn dimensionless numbers.  “Probability waves are empty in that they carry no energy or mass.  Numerical relations are their exclusive contents.”  “At the foundation of reality, we find numerical relations -- non-material principles -- on which the order of the universe is based.” [1]  

But it doesn’t end there.  Quantum “stuff” is basically “mind-stuff” -- see Wave-Particle Duality and/or Quantum Knowing.  “The mind-like properties of the background of reality are also suggested by the fact that its order is determined by principles of symmetry [aka geometry], abstract mathematical patterns, to which the constituents of the material world have to conform.” [1]  

Schäfer also references Werner Heisenberg -- one of the most known world class physicist (for whom the Heisenberg Uncertainty Principle is named). “The elementary particles in Plato’s Timaeus are not substance but mathematical forms.  ‘All things are numbers’ is a sentence attributed to Pythagoras.  The only mathematical forms available at that time were such geometric forms as the regular solids or the triangles which form their surface.  In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms, but of a much more complicated nature.” [1] [emphasis added]  And maybe, just maybe, not that much more complicated!  

In sum: “In the quantum phenomena, we have discovered that reality is different from what we thought it was.  Visible order and permanence are based on chaos and transitory entities.  Mental principles -- numerical relations, mathematical forms, principles of symmetry -- are the foundations of order in the universe, whose mind-like properties are further established by the fact that changes in information can act, without any direct physical intervention, as causal agents in observable changes in quantum states.” [1]  

Calitatile numerelor

1...2....3...4...5...6...7...8...9...si APOLLON.

THE MOUSAI (Muses) were the goddesses of music, song and dance, and the source of inspiration to poets. They were also goddesses of knowledge, who remembered all things that had come to pass. Later the Mousai were assigned specific artistic spheres: Kalliope (Calliope), epic poetry; Kleio (Clio), history; Ourania (Urania), astronomy; Thaleia (Thalia), comedy; Melpomene, tragedy; Polymnia (Polyhymnia), religious hymns; Erato, erotic poetry; Euterpe, lyric poetry; and Terpsikhore (Terpsichore), choral song and dance.

In ancient Greek vase painting the Mousai were depicted as beautiful young women with a variety of musical intruments. In later art each of the nine was assigned her own distinctive attribute.

There were two alternative sets of Mousai--the three or four Mousai Titanides and the three Mousai Apollonides.

.

Numere perfecte (Teleioi) si ciurul Teleman

Numerele perfecte sunt plasate doar in doua spatii din ciurul Teleman,.Locurile acestea sunt evidentiate prin sageti.
Pentru cautarea numerelor perfecte am gasit o alta metoda.
Cu regula lui Euclid ,foarte restrictiva ,se considera doar numerele ''perfect'' de perfecte,unde rolul determinant il au numerele prime nontriviale.
Metoda mea valorifica si numerele prime triviale..

.........
PROPERTIES OF MERSENNE NUMBERS AND PRIMES
If one looks at the sequence of numbers-
M(p)= 3, 7, 31, 127, 2047 , 8291, 131071, 524287
one notices that its elements are, with the exception of 2047, prime numbers
defined by –
M( p)  2 p 1 with p being the primes 2, 3, 5,7, 11, 13,...
These numbers for any prime p are known as Mersenne Numbers named after the
French cleric and mathematician Marin Mersenne(1588-1648). To date less than
fifty of these numbers have found to be prime. The vast majority of the numbers
M(p) are composite numbers especially at large p. It is our purpose here to
examine these Mersenne Numbers in more detail.
The first thing we notice is that they are always odd numbers when p3 ending in
either 1 or 7 . This means that –
M(p) mod(6) =1
whenever p3 without exception. It implies that Mersenne Numbers lie along the
radial line 6n+1 in the following hexagonal integer spiral diagramThe
M(p) primes are quite sparse compared to the Q Primes along this same radial
line of 6n+1. Only M(p)= 7 and 31 are shown in the diagram. Now 7 and 31 differ
from each other by 24= 6x4. The next gap for M(p) will be 127-
31=96=6x16=12x8=24x4=48x2. These observations suggests that we might
identify Mersenne Numbers by mod values mod(6) followed by mod(12),
mod(24), etc. We have done so and get the following resultsp
M(p) M(p)
mod(6)
M(p)
mod(12)
M(p)
mod(24)
M(p)
Mod(48)
M(p)
Mod(96)
2 3 - - - - -
3 7 1 - - - -
5 31 1 7 7 - -
7 127 1 7 7 31 31
11 2047 1 7 7 31 31
13 8191 1 7 7 31 31
17 131971 1 7 7 31 31
19 524287 1 7 7 31 31
23 8388607 1 7 7 31 31
29 536870911 1 7 7 31 31
31 2147483647 1 7 7 31 31
The table shows that the mod operations using 6, 12 ,24, 48, 96, etc yields the
same constant number for a fixed mod. The mod sequence reads-
1, 7, 31, 127, 511, 2047, 8191, …
where the elements are recognized as–
22m1 1 with m  0, 1, 2, 3, 4, 5,...
For mod(6x2m), the Mersenne Prime will lie along only the single radial line
(6x2n)+const , where the constant represents the appropriate constant for a given
mod. This means, for example , that M(p)=127 will lie along the radial line
48(k)+31 at k=2 in an integer spiral with 48 radial lines. The next Mersenne
Number along this line will be 48( k)+31=2047 with k=42.
In the above table the Merseene Numbers corresponding to p=11, 23, and 29 are
composite, while the others are primes. The mod operations do not distinguish
between the two types of numbers, namely, composite or prime. This distinction
must be found by some type of primality test. The simplest of such tests is that of
Fermat known as Fermat’s Little Theorem. It states that a number is prime if-

 
p
(a p 1 a)
integer
, where a is a low integer such as 2, 3, 4,… . One can simplify this result by
recasting things using modular arithmetic. Its equivalent form reads-
(ap-1-1)mod(p)=0
So if p=127, we get-
(2126 -1)mod(127)=0
, meaning 127 is a prime. What about the Mersenne Number M(11)=2047? Here
we get-
(32046-1)mod(2047)=1012
indicating that M(11) is a composite. Note that here using a=2 would not work as
a test.
Since in modular arithmetic we have the identity-
(AxBxC) mod(N)=[A mod(N)]x[B mod(N)]x[C mod(N)]
we can simplify the Fermat test for primeness by writing-
(a p1 1)mod(p)  [(a( p1) / 2 1)mod(p)]x[(a( p1) / 2 1)mod(p)]
Sometimes this type of breakup can be continued for several more terms making
the mod operations a lot easier. Take the Mersenne Number 8191. To test if it is
prime we write-
[(2 1)mod(8191)] [(2 1)mod(8191)] [0] [2] 0
(2 1)mod(8191)
4095 4095
8190
   
 
x x
So it is a Mersenne Prime. To make super-sure, we can replace 2 by any other
positive integer. Doing so, we find-
(38190 1)mod(8191)  0
and the primeness of M(13) is confirmed.
A question often asked is how did Mersenne come up with the form for M(p). The
answer is that he was studying the works of the ancient Greek mathematician
Euclid. In Euclid’s book it is shown that-
1+2=3
1+2+4=7
1+2+4+8=15
1+2+4+8+16=31
1+2+4+8+16+32=63
1+2+4+8+16+32+64=127
or, in general, that-
2 2 1 1
0
  
 
N
N
n
n
Now, if N+1=p, then the right hand side of this last expression just represents the
Mersenne Number M(p). It yields the identity-
2 2 1 ( )
1
0
p M p
p
n
n    


This is most likely the way Mersenne came up with his number. It says, for
example, that –
M(7)=1+2+4+8+16+32+64=127
It should be pointed out that the Mersenne Numbers are not the only numbers
capable of producing primes via the above approach. By generalizing the above 2n
series, we have-
( 1) 2, 3, 4, 5,... 1
0




 
where a
N
a a
N N
m
n
If we now take the odd number a=3, then aN+1 will also be odd. Since prime
numbers above p=2 are also odd numbers, it suggests one define a generalized
Mersenne number sequence given by-
F(a,N,b)=aN  b. with a  2,3,4,5,..and b  1,2,3,...
In an integral spiral with six radial lines the function F(a,N,b) will lie either along
the line 6N+1 or 6N-1. The corresponding mod operation will yield 1 or 5.
We have played around with various combinations of a and b and find one of the
richest forms for yielding primes is-
F(3,N,2)=K(N)
Here K(N) mod(6)=5 meaning these numbers all lie along the radial line 6N-1 in
the above integral spiral. We have used our PC to find the values of N for which
K(N) is a prime. Here are the results when running over the range 0<N<2000 :
N=1,2,3,4,8.10.14,15,24,26,36,63,98,110,123,126,,139,235,243,315,363,386,391,
494,1131,1220,1503,1858,
This shows a total of 28 primes in the range considered , making the numbers
K(N) more dense in primes than the standard Mersenne Primes of which less than
50 have been found to date. Here is the prime number corresponding to K(1858) :
31858+2=30994973482657325447985147127620089298530431006956594685920
214825663326619113769057695213547484440453837246123010100669127100
955759919127457869223148660164613467627385033860708738672704316650
862181089633382991315337701542754751372995001149340588749380743536
244008880419755202390383627639285895496613970962701489924969795727
825590048078397355734118492863593725501268400567250135296630863031
128376511219475662720257626805673311452744631321737696232779367051
782399514104280263667180701547531032088649276210976147452750560033
145878420041510221647182900875021810238965863623224904713399693521
685908522220999474927503580442427935550946200830198700373449243582
744499189296500580263454833210298365535155326181801482295213278862
293406357314198353241285393759520549296816928599853248072473293309
376537937390628464770980014113878906045022870019937434926857282944
448276114607243265213141322025447691
This 790 digit long prime has, as expected, the value K(1858) mod(6)=5.
We have also played around with other values of ‘a’ and ‘b’. None are found to
be quite as rich in primes as the K(N) numbers.
An additional generalized Mersenne Number sequence which we examined in
some detail is-
F(2,N,1)=2N+1
Searching this function for primes, we were able to find only the five primes
given in the following table-
N F(2,N,1)
1 3
2 5
4 17
8 257
16 65537
In looking at the progressions of N in the table it is clear that they are given by
N=2n. That is, the number F(2,N,1) can also be written as-
F(2,2n,1)=22 1 n
But this function is recognized at once as representing the Fermat Numbers.
Fermat originally thought all these numbers are prime for any positive integer N,
but Euler proved him wrong by showing that 232+1 factors into
4294967297=641 x 6700417
Since that time no additional Fermat Primes have been found, so it is safe to say
that F(2,2n,1) are all composite numbers when n5. Also we have searched
F(2,N,1) over the entire range 16<N<1000 and find no additional primes. It leads
to the conjecture that –
The infinite sequence of numbers F(2,N,1) contains only five primes
corresponding to N=1,2,4,8,16 and no more.
Finally we point out that there are an infinite number of other number sequences
which have a much higher prime number density then the Mersenne Numbers.
One of these which comes to mind is G(N)=N2+(N1). It has a total number of 83
primes out of the first 200 possibilities ( 1<N<100 for both + and – case).
U.H.Kurzweg
Septemver 18, 2015

The trivial prime numbers and Teleman sieve.

The trivial prime numbers and Teleman sieve.

Сырьевые числа и Телеман тривиальное сито.

Numerele prime triviale si ciurul Teleman.

.

A prime trivial number is a product of nontrivial prime numbers.

Сырой продукт не является тривиальным числом нетривиальных простых чисел.

Un numar prim trivial este un produs de numere prime nontriviale.

.

Tabel cu numere prime non triviale.

Table of non trivial prime numbers.

Таблица нетривиальных исходных чисел.

.

Exista ordine in sirul numerelor prime triviale.

Iata un exemplu .

Numerele au doua componente ,o parte fixa ;desenata cu culoarea rosie ,si o parte variabila ,desenata cu culoarea neagra.

There is order in the trivial prime numbers.

Here's an example.

The numbers have two components, a fixed part, drawn in red, and a variable part, drawn in black.

. В тривиальных простых числах есть порядок.

Вот пример.

Числа содержат две компоненты: фиксированную часть, красную и переменную часть, черную.

.

De exemplu,in acest tabel,folosesc o regula simpla pentru a descoperi numerele prime triviale,folosesc adunarea cifrelor ,scrise cu culoare neagra .

For example, in this table, I use a simple rule to find the trivial prime numbers, using the digits, written in black.

Например, в этой таблице я использую простое правило для поиска тривиальных простых чисел, используя цифры, написанные черным цветом.

.

Parole de la turnul Babel

.

Sieve Teleman prime number

Prime numbers ,Zodiac ,Pitagora

Prime numbers

O idee de un milion_?

Teorema Teleman ,numere prime,prim numbers

Prime numbers ,numere fundamentale

.

Tarot ,Oul Matematic ,si Teorema Teleman

Tarot ,Oul Matematic ,si Teorema Teleman (gandirea ''gordianica'')