Brief Summary of and Addendum to Number Theory
and the Avta Foundation for Primes
version: one point one
date: 24 February 2003
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In 2001, I published a monograph on Number Theory and the Avta Foundation
for Primes at http://www.ebookmall.com/ebook/29054-ebook.htm,
based on an infinite set of numbers beginning with 1, 7, 11, 13, 17, 19, 23, 29, that I discovered in 1991 and eventually called the Avta
Number System (ANS), or the Avta Foundation for Primes. Avta was a term I derived from an old Norse
word atta meaning eight to represent
the eight elements in every ANS cycle [Rask,
Erasmus. A Grammar
of the Icelandic or Old Norse Tongue. Translated from the Swedish
by George Webbe Dasent,
M.A. William Pickering:
Recently I realized, in examining a university course write up on Discrete Mathematics and a textbook on the subject, that every simple English word that might be used to describe a collection of numbers, such as system, group, series...already has a very specific meaning in mathematics. Therefore, my innocent reference to Avta, Chi, and other number groupings in my monograph as "Number Systems" was probably not acceptable to at least some professional mathematicians. I should have referred to these number groupings as "sets" and at least have supplemented my view of natural numbers, and some of the different possible sets within natural numbers, by Venn diagrams. In addition, I learned from Ed Page of Wolfram Associates that Coxeter and Moser, had referred to the same set of numbers in Generators and Relations for Discrete Groups published in 1984. Upon checking the book, I found only a two-line reference in Table 10, page 141, to a finite group consisting of 696,729,600 elements beginning with the same eight integers. The book was actually a 4th edition, and the 2nd edition published in 1955, which I was also able to obtain, had only the same two line reference, but no suggested relationship to prime numbers, to the infinite virtually composite number sets, one for each element, to the basis for prime twins and other tuples, and to many other important features of this amazing set of numbers as discussed in my monograph.
To briefly recapitulate my publication, which is still available on eBookMall, the set of natural numbers in which all primes except 2, 3, and 5 occur, and which I called the Avta Number System in my book, I should now refer to as the Avta Number System composed of an infinite number of Avta Number Sets, still abbreviated ANS, all ending in 1, 3, 7, and 9 in base 10. The elements in these sets follow specific repetitive linear addition patterns in their eight-membered cycles, of +6a, +4a, +2a, +4a, +2a, +4a, +6a, in each 30a consecutive natural numbers, followed by an eighth addition, a +2a bridge, to the first element in the next cycle. In this addition pattern, a is equal to the initial avta in a set, each set is named for its initial avta, and referred to as a wheel. And the eight elements in each cycle of a wheel are referenced by their position in a cycle from the first to the eighth as Positions 1 through 8, or P1, P2, …P8. The universal avta set called the Primary Wheel or Wheel 1 begins with the number 1 considered prime, includes all the other ANS sets, contains all primes greater than 5, but is not exclusively prime, because it includes non-prime multiples of earlier elements, but not multiples of 2, 3, or 5.
An unknown natural number ending in 1, 3, 7, or 9 is an avta and may be prime if when divided by 3 it has a remainder of either 1 or 2. Such an unknown number is also an avta if when divided by 30 it has a remainder equal to one of the eight elements in the first cycle of Wheel 1, namely 1, 7, 11, 13, 17, 19, 23, 29, and its position will coincide with the position the remainder held in that first cycle. Quotients with no remainder, resulting when numbers such as 30, 60, or 90 are divided by 30, indicate a transitional boundary point between cycles. Division by 3 is merely a quick method of identifying an avta but gives no indication of the avta’s position in its cycle. An avta cycle is equal to the whole number portion of the avta when divided by 30 plus 1. For example, the number 17 when divided by 30 has a quotient of 0 and a remainder of 17. Seventeen is therefore an avta in Cycle 1 (0 + 1 = 1), is in Position 5 in that cycle, and can be represented by C1P5. To find an avta’s location in the ANS, multiply the whole number portion of the quotient of the avta when divided by 30 by 8 and add the position value of the avta in question. Ergo, find the nth avta location of 37: when 37 is divided by 30, the quotient is 1 and the remainder is 7. Since 7 is a Position 2 avta, 8 x 1 + 2 = 10; that is, 37 is the 10th number in Wheel 1 of the Avta Number Set, or it occupies the 10th location in that set.
Wheel 1 can be satisfied by either the formula y = 30x –v, where y equals a Wheel 1 avta, x equals the cycle number in which the avta occurs, and v is a vector consisting of all elements in the first cycle of Wheel 1 arranged in reverse order from 29 to 1; or, y=30x + v, where x equals the counting numbers beginning with 0, and x + 1 would equal the cycle number. The values for v in both formulae would be the same, but in the +v formula the order of the elements would be from 1 to 29.
The higher wheels, beginning with 7, 11, 13…, one wheel for each avta,
create the almost-prime composites (my terminology) in the universal
wheel at points of intersection with that wheel, accounting for all
non-primes/composites ending in 1, 3, 7, and 9 in Wheel 1 of the ANS, and
together with the Limited 3 Set discussed later in this article account for all
non-primes/composites ending in 1, 3, 7, 9 in the Set of Natural Numbers. All wheels ending in 1, such as Wheels 1, 11,
31, 41, have in each cycle 1, 7, 1, 3, 7, 9, 3, 9 as their end-digit
pattern; those ending in 3 have 3, 1, 3, 9, 1, 7, 9, 7; in 7,
7, 9, 7, 1, 9, 3, 1, 3; and, those wheels ending in 9 have 9,
3, 9, 7, 3, 1, 7, 1. One hundred and twenty cycles of the first ten
higher wheels, i.e. Wheels 7 through 41, are given in my monograph, and
represent, I believe, the initial 120x8 or 960 elements in the first ten of an
infinite number of new number sequences.
The wheels within wheels thus occurring, with a maze of interactions
between the wheels, can I believe be called the Avta
Number System, basically a recursive system, with each wheel in the system
forming one set. Wheel 7, for example,
in one revolution or cycle will roll over 7x30 or 210 natural numbers, striking
eight avtas in seven cycles of Wheel 1, beginning
with the first Wheel 1 cycle. Seven is
the only higher wheel that strikes one avta in each cycle and two in the fourth
of every seven cycles. The avtas struck are non-primes divisible by 7, and equate to
at least 14.29% of the ANS. The
location, cycle, and position pattern for these higher wheels are
characteristic for each wheel, and can be used as a sieve of Eratosthenes to
produce a database of primes, or as Sir Arthur S. Eddington
might say: to remove the sand from the desert
and leave the lions [taken from David M. Bressoud’s Factorization
and Primality Testing p. 102, Springer-Verlag, New York, c. 1989.
From an examination of my tables for Wheel 1 and the higher wheels in Appendix A of my monograph, in which tables zero represents a prime and one a composite, I have come to the conclusion that the concept and definition of primality and the large body of conventional primes as currently understood are based upon the numbers 2, 3, 5, and Wheel 1 of the ANS, in which a prime is defined as being divisible only by one, that is by the initial avta in Wheel 1, and by the number itself. Since all avtas are only divisible by themselves and earlier elements in their wheel, all primes in Wheel 1, that is the controversial number 1 and all primes greater than 5, are numbers that except for being divisible by the initial avta 1, are only divisible by the avta itself and not divisible by other earlier avtas in the primary wheel. Based on a similar rule, I propose each higher wheel has its own elements that are prime and others that are composite with respect to that wheel. Thus, a “prime” in a higher wheel, as in Wheel 1, is only divisible by the initial avta in the wheel and by the number itself, but not by earlier elements in that wheel. In Wheel 7 for example, a prime should only be divisible by 7 and by the number itself—not by earlier elements in the 7 Wheel; in Wheel 11, by 11 and the number itself, but not by earlier elements in the 11 Wheel; et al. Composites in each wheel begin at the square of the Position 2 number in the first cycle of the wheel, and follow a pattern applicable to all wheels, namely: P2 x P2, P2 x P3. P2 x P4, P2 x P5, P3 x P3, P2 x P6, P3 x P4, P2 x P7, P4 x P4…., which in Wheel 1 corresponds numerically to 7x7, 7x11, 7x13, 7x17, 11x11, 7x19, 11x13, 7x23, 13x13… The order can be compared with Table Vb on p. 730 of the VNR Encyclopedia 1977. As a result, most of the numbers in the first 120 cycles of each higher wheel are prime for their specific wheel. Primality in my opinion is therefore a relative term with respect to the Avta Number System since the ANS consists of an infinite number of sets. In this case, primality of a number in a set is dependent upon the wheel or set to which it belongs and could be redefined as: a number which is only divisible by the initial number in its specific wheel or set in the ANS, and by the number itself, but not by other earlier elements in that wheel or sequence. However, all numbers that are prime with respect to a specific higher wheel of the ANS remain composite with respect to Wheel 1, except the initial avta in the wheel, as the initial avta may or may not be prime in Wheel 1; for example, Wheels 7, 11, 13 begin with conventional primes, Wheels 49, 77, 91 begin with non-primes.
The Coxeter set or group beginning with 1, 7, 11, 13, 17, 19, 23, 29 [Coxeter and Moser, Generators and Relations, 4th Edition 1984, Table 10, p.141] to which I was referred by Ed Pegg Jr. of Stephen Wolfram Science Group, on January 6, of this year, is a finite set consisting of some 600 to 700 million elements. Wheel 1 of the ANS could somehow I suppose be considered an extension of Coxeter’s set to infinity with an expansion of its significance. Wheel 1, the 1-7-11 Set of the ANS is infinite, reintroduces the number 1 as a prime, contains all primes greater than 5, each cycle in any set contains eight elements, the linear additive arrangement of these elements in each cycle explains the existence of prime twins, triples, quadruples, tuples, quadruplets, and sextuplets; no element in this Set or any other higher wheel in the System is divisible by 2, 3, or 5, and the occurrence of higher wheels explains the existence of non-primes in the ANS, and hence many of the non-primes in the set of Natural Numbers since the ANS is a subset of the natural numbers.
Lack of divisibility of members of the ANS by 2, 3, and 5 can be explained based upon the fact that these numbers are not elements of the ANS, and members of the ANS are only divisible by earlier elements in Wheel 1 of that system. The Chi Number System/Set on the other hand contains these three conventionally accepted primes 2, 3, and 5 but all other elements in the Chi Number Set are composite, divisible by 2, 3, and/or 5, or by one or more avtas. The other number groupings mentioned in my monograph, such as the well known 3-5 Number Set, the 2 Number Set,… further expanded here with Venn diagrams and relevant discussion, are self-explanatory.
In a universal Venn diagram with one set made up of subset A and its complement A' (called “A prime”) is shown below:
1) If the universal set U is the set of all natural numbers, then the 1-7-11 or Wheel 1 of the Avta Number Set, is subset A of the natural numbers, and the 2-3-5 or Chi Number Set is the complement of the Avta Numbers, or A' (Section 1.1 The Conceptually New Subdivisions of the Natural Number Set, p. 5-10 of monograph). To simplify this presentation, in every group of 30 consecutive natural numbers, eight are members of Wheel 1 of the ANS and 22 numbers belong to the CNS. Both of these sets are infinite, each with a specific and identifying addition pattern sequence. The first eight numbers in Wheel 1 of the Avta Number Set are 1, 7, 11, 13, 17, 19, 23, and 29, separated in each 30 consecutive natural numbers by the addition sequence +6, +4, +2, +4, +2, +4, +6, with a +2 bridge to the first element, 31, in the next avta cycle. The eight numbers in Cycle 2 are therefore 31, 37, 41, 43, 47, 49, 53, and 59. The additive arrangement of the elements in this set provides the underlying basis for the distribution of all primes greater than 5, including prime twins, triples, quadruples, and tuples, as well as prime quadruplets and sextuplets (Section 2.3 Prime Terminology: Twins, Quadruplets, Sextuplets, Triples, Quadruples, and Tuples, p. 27-29 of monograph). All historical number and formulae associated with primes in the past, present, or future—such as Fermat, Mersenne, Fibonacci, Lucas, Carmichael Numbers and Prime-Producing Polynomials--will contain, produce, or be associated with avtas (Section 1.3 Historical Number Systems and Formulae p. 13-18 and Appendices B-F p. 125-143). The remaining 22 numbers from 2, 3, 4, 5, 6, 8, 9, 10, 12,14, 15... to 30 belong to the Chi Number Set . Chi is the 22nd letter of the Greek alphabet, hence the set name Chi was chosen to represent the 22 elements in each Chi cycle in 30 consecutive natural numbers.
2) If the universal set U is the set of all odd natural numbers, then the ANS is subset A of the odd natural numbers, and the 3-5 Number Set is the complement of the ANS, or A'. The 3-5 Number Set was used as one of the possible complements of the ANS in my monograph because these two sets, the ANS together with the 3-5 Number Set equate to the set of all odd natural numbers. In each consecutive group of 30 natural numbers beginning with the number 1, 15 are odd and 15 are even. Of the 15 odd, eight belong to the ANS and 7 belong to the 3-5 Number Set. All 15 even numbers belong to the 2 Number Set. The first eight numbers in the Avta Number Set have already been given. The first 7 numbers in the 3-5 Number Set are 3, 5, 9, 15, 21, 25, 27, separated in each 30 consecutive natural numbers by +2, +4, +6, +6, +4, +2, with a +6 bridge to the first element, 33, in the next 3-5 cycle.
3) Since all primes except the number
2 are odd numbers that end in 1, 3, 7, or 9, if the universal set U is the set
of all odd numbers ending in 1, 3, 7, and 9, then the Avta Number Set, A, is a
subset of U and the elements of Set 3 that end in 1, 3, 7, and 9 are the
complement of the Avta Number Set, represented by A'. There are only 12 odd numbers in every 30
consecutive natural numbers that end in 1, 3, 7, or 9. Eight of these numbers
belong to the Avta Number Set and four belong to Set 3 or A' just described,
which was not separated from the 3-5 Number Set in my monograph, but could have
been easily separated from that set by removing the numbers 5, 15, and 25. The first four elements of the Set of 3, that
is limited to numbers ending in 1, 3, 7, or 9, are 3, 9, 21, 27,
separated by +6, +12, +6, with a +6 bridge to the first element, 33, in the
next cycle. The combined composites in the Avta and the Limited 3 Set provide
the reason for the apparent random occurrence of primes, and the large gaps
that occur between primes further down the natural number road. As Ribenboim astutely noted in 1996, the combination of short
interval randomness among primes along with the predictability of the number of
primes smaller than N, especially if N is large, indicate an orderly
arrangement in the distribution of primes [Ribenboim,
Paulo. The New Book of Prime Number Records.
Springer-Verlag:
4) If the universal set U is the set of all Avta numbers, then Set A is a subset of U containing all prime avtas, that is all primes except 2, 3, and 5, and the complement A' contains all non-prime, or as I like to call them, almost-prime avtas, resulting from the multiplication of earlier elements in the ANS, such as 7x7, or 49, and 7x11 or 77.
In each of the preceding four cases, A and A' form infinite subsets of U.
Or, in a Venn diagram with three sets/subsets, A, B, C, all of which are infinite sets that can unite to form the 2-3-5 or Chi Number Set, the complement to these three subsets is (ABC).’ However, since infinite sets could not be shown numerically, the numbers included in this diagram are taken from only the first 30 consecutive natural numbers to help elucidate with a specific example how these sets overlap.
Ignoring the specific numbers included in this diagram
If Set A = the set of all even natural numbers divisible by 2, that is the 2 Number Set;
Set B = the set of all natural numbers divisible by 3, that is the 3 Number Set;
Set C = the set of all natural numbers divisible by 5, that is the 5 Number Set;
then Set (ABC)' will be the set of all odd natural numbers not divisible by 2, 3, or 5, namely the Avta Number Set.
In other words, the numbers 2, 3, and 5 are conventionally accepted primes but flagships of three totally composite and overlapping number sets. Since 2x3x5 = 30, the 2 Number Set overlaps the 3 and 5 sets every 30 natural numbers, at 30, 60, 90, 120...to infinity. Since 3x5 = 15, the 3 and 5 Number Sets overlap every 15 natural numbers, at 15, 30, 45, 60, 75, 90....
The example of a Venn diagram supposedly showing the relationship between
Fibonacci, Odd, and Prime Number subsets in the set of natural numbers, given
on p. 493 of Microsoft’s Computer Dictionary, c. 1977, is incorrect and
shows a lack of understanding of Fibonacci’s. As explained in Appendix D of my monograph,
the Fibonacci Numbers are arranged in recurring sequences of 60 numbers, each
sequence beginning with numbers that end in 0, 1, 1, 2, 3, 5, 8….[Jarden, Dov
“Table of Fibonacci and Lucas Numbers” in Recurring Sequences, A Collection
of Papers, 2nd Ed Revised and enlarged with numerous new
factorizations by John Brillhart. Riveon Lematematika,
with the tiny subset A representing the Fibonacci Numbers, B representing
the set of Odd Natural Numbers, and C,
the Prime Number Set composed of prime avtas plus 2,
3, and 5, which are not only conventional primes but also prime Fibonacci’s. ABC’
would be the set of Even Natural Numbers less the even Fibonacci’s. Since the Fibonacci subset is so very small,
probably not more than a dot on the edge of the Odd- Number subset circle, a
recognizably accurate diagram would be difficult to produce. Fibonacci’s are
apparently not a set because they contain one duplicate number, the number 1,
in the first sequence [Breuer, Joseph Introduction
to the Theory of Sets. Translated by Howard F. Fehr. Prentice-Hall:
Similar Venn diagrams could be created for other famous number systems associated with primes.
As suggested by Lehmer in 1913/1956, since the
number 2 is the only even number that is considered prime, it should probably
be excluded from the list of primes [Lehmer, Derrick
Norman. List of Prime Numbers from 1 to 10,006,721.
Carnegie Institution of Washington, Publication No. 165, with
a forward by D.H. Lehmer, 1913. Hafner Publishing:
As mentioned, the Avta Number Set (ANS) may also be considered to consist of an infinite number of sets, one for each avta, which can be represented by scalable eight-spoked wheels, one spoke for each of the eight elements in one cycle or revolution of any wheel, with the wheel named for its initial avta (Section 3. Graphical Representations of the Avta Number System, p. 53-55). The external numbers 1 to 8 in the non-artistic black and white clock-face diagram shown here refer to the eight unchanging positions held by the avtas in each cycle. The numbers 6a, 4a, 2a, with “a” equal to the initial avta in the wheel, are the numeric distances, or addition sequence pattern, between these numbers in each cycle, and rapidly become huge as the wheel numbers increase.
Graphical Representation of a
scalable eight-spoked Avta Wheel
Wheel 1 in this representation is the universal set of Avta Numbers, containing all prime and non-prime avtas (Sections 2.1 and 2.2 p. 21-26, and Appendix A First 120 Cycles of Wheel 1…. p. 88-94 of monograph). In Wheel 1, since “a” is 1, 6a, 4a, 2a equal 6x1, 4x1, 2x1, or 6, 4, and 2. The addition of 2a to the eighth number in an avta cycle will give the first number in the next cycle of any Avta Set. In Wheel 1: 29 + 2 = 31, the Position 1 avta in Cycle 2. The clock-face wheel above is numbered counterclockwise because as the wheel rolls over a number line, the spokes of the wheel will strike positions in an increasing order from one to eight. If the wheel were labeled clockwise, the positions struck would be in descending order from eight to one.
To repeat my earlier statements, non-prime avtas are created by the intersection of higher wheels of the ANS with Wheel 1: higher wheels such as Wheels 7, 11, 13, 17 (see p. 30-36 and Appendix A The First 120 Cycles of [Wheel 7] through Wheel 41… p. 95-124 of my book). Higher wheels are subsets of Wheel 1 and of each other. Thus the Avta Number System can also be considered to consist of an infinite number of wheels within wheels, or a logical identity. By the term logical identity I mean there is a pattern underlying the operation, functioning, or behavior of the identity.
The two colored avta wheels shown below, with the distance 2a represented by white, 4a by red or purple, and 6a by blue or yellow, are more artistic renderings of a cycle in the ANS, capable of conversion to a small hanging pendant, a 3D crystal cut, or a stained glass window of any size, and emphasize the symmetry existing in the Avta Number Set. When referring to these wheels, I call the red, white, and blue one: Patriotic Colors, and the yellow and purple, Easter Sunrise. Any three or even four colors could have been used. There are only three different number differences, but a fourth color could be applied to the gap at the midpoint as I initially had difficulty determining whether to break the first cycle of Wheel 1 at 15 or 30. The avta wheel shown on page 54 of my monograph is a more complex artistic representation of the ANS but may be confusing because it combines in one diagram the ANS circle representation on p. 56 with the wheel, and the diagram colors on p. 54 have no significance other than to the program I wrote to generate them; whereas, the colors used in the Patriotic Colors and Easter Sunrise Wheels connote and emphasize the addition sequence pattern (refer to the corresponding black and white wheel diagram above). I hope my simplified explanation of number theory and the foundation for primes will make it possible for even grade school students to understand these concepts.
Patriotic Colors Wheel Easter
I was recently surprised to learn that my spoked-wheel
representation of the Avta Number Set has a name, the mandala,
circular in shape with spokes radiating from the center, and is found in
nature, in cave paintings, the Aztec calendar, and in the early art of most
societies--in Hinduism, Buddhism, and Sufism, being used as an aid to
meditation [Errol R. Korn, and Karen Johnson Visualization,
the uses of Imagery in the Health Professions, Chapter 5 Imagery, pp.73-74,
and in various sections of the Encyclopedia Britannica. Thus my supposed original insight into and
understanding of the ANS as represented by eight-spoked
wheels rolling along the natural number road to infinity supports Carl Jung's
observation in 1933, that images are representations of deep inner
experiences and that similar images will surface in different cultures when
they are needed to guide the way or solve a problem [Pratt, Wood, Alman. A Clinical Hypnosis Primer.
Wiley:
Geometric Avta Tile Example
Arranged by Hand
