As an addendum to my monograph on Number Theory and the Avta Foundation for Primes, I have recently realized, in examining a course write up on Discrete Mathematics and a textbook on the subject, that my innocent reference to Avta, Chi, and other number

Addendum to Number Theory and the Avta Foundation for Primes

version one

In 2001, I published a monograph on Number Theory and the Avta Foundation for Primes at http://www.ebookmall.com/ebooks/showdetl.cfm?&DID=8&Product_ID=29054. 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 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.

To briefly recapitulate my publication, which is still available on eBookMall, the newly identified 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 now call the Avta Number Set, still abbreviated ANS. This set is infinite but not exclusively prime, because it includes non-prime multiples of earlier elements. Since the numbers 2, 3, and 5 are not elements of the Avta Number Set, no Avta is divisible by 2, 3, or 5. The Chi Number System/Set on the other hand contains the three conventionally accepted primes 2, 3, and 5 but all other elements in this set are composite, divisible by 2, 3, and/or 5, or one or more avtas. The other number groupings mentioned in my monograph, such as the 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 Avta Number Set, the conceptually new number set referred to in my recapitulation, 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 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 the Avta Number Set are 1, 7, 11, 13, 17, 19, 23, and 29, separated in each 30 consecutive natural numbers by the addition pattern sequence +6, +4, +2, +4, +2, +4, +6, with a +2 bridge to the first element, 31 in the next avta cycle. 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 . Avta is a term I derived from an old Norse word atta meaning eight (Rask 1843), referring to the eight elements in each ANS cycle occurring in 30 consecutive natural numbers, and Chi is the 22^nd letter of the Greek alphabet, the set name being 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, 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 and 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, 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: New York, 1996, p. 213]. Except for 2, 3, and 5, the Avta Number Set provides this orderly arrangement or foundation for all other primes.

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, 2^nd Ed Revised and enlarged with numerous new factorizations by John Brillhart. Riveon Lematematika, Jerusalem (Israel), 1966. pp 40-64 of 137 pages]. Each cycle of 60 Fibonacci’s has 20 even numbers ending in 0, 2, 4, 6, 8; and 40 ending in the odd numbers 1, 3, 5, 7, 9; that is, 1/3 of all Fibonacci’s are even, and 2/3 are odd. Of the 40 odd numbers, 24 or 40% are avtas, some of which are prime. In the first Fibonacci sequence, there are only 60 Fibonacci’s in almost 957 billion natural numbers, a very, very small group indeed, and 13 of the 60 are prime—the numbers 2, 3, and 5 and ten avtas, namely 1, 1, 13, 89, 233, 1597, 28657, 514229, 433494437, and 2971215073. Thus Fibonacci’s are a miniscule subset of the Natural Numbers Set, odd numbers form a 50% subset of the natural numbers, and avtas form a 26.7% subset. Of the avtas only a small percentage are prime, and an even smaller percentage are also prime Fibonacci’s. Pomerance, In Search of Prime Numbers [Sci. Amer. 247 (6): 136-144, 1982] states Jan Bohman of the University of Lund has shown there are only 882,206,716 primes in 20 billion natural numbers, which equates to just 4.5% primes or prime avtas, 95.5% total composites, and 22.16% non-prime avtas (Section 2.10 ANS Percentages in the Natural Number Set, p. 47-48). A more correct Venn diagram of Fibonacci Numbers, Odd, and Prime Number Sets including the number 2, would therefore be:

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: Englewood Cliffs, NJ, c. 1958, 5^th Printing 1963. p 6 of 108 pages] and can only be referred to as a sequence since in a sequence an element may be chosen more than once, and the sequence of Fibonacci numbers is defined by a₁ = 0 [[VNR Concise Encyclopedia of Mathematics referred to before, p 381].

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: New York, c. 1956, p. 1]. Zero is essentially part of the 2 Number Set, separating the positive and negative halves of that system, and from this standpoint can be considered an even number (my belief derived from the simple interleaving pattern of odd/even numbers and the pattern association with 30 as the even number between Avta Cycles). The number 1 has been considered a prime by some mathematicians and not by others, for one reason because the conventional definition of a prime is a number divisible by 1 and by the number itself, and in the case of the number 1, it has only one divisor, 1, and this is also itself [Rózsa Péter, Playing with Infinity: Mathematical Explorations and Excursions translated from the Hungarian by Dr. Z.P. Dienes first published in English in 1961, and republished by Dover Books in 1976, p.57]. The number 1 is the first number in the Avta Number Set and I join once again with Lehmer in classifying it as a prime. I do not agree with the statement on p. 24 of the VNR Concise Encyclopedia of Mathematics, with Gellert, Kustner, Hellwich, and Kastner as editors and Hirsch and Reichardt as Scientific Advisors, Van Nostrand Reinhold Company, New York 1977, that Euclid’s algorithm would be incorrect if 1 were counted among the primes. Using their example, but with 1 included as a prime, the number, 1008, when factored into primes would now equal 1.2.2.2.2.3.3.7 or 1 .2⁴ .3². 7. The number 1 does not add unique information to the list of prime factors, however before a prime number is classified as non-prime for the sake of convenience, someone should rethink the wording of the proof for Euclid’s algorithm, which does an efficient job of finding the greatest common divisor between two numbers. I also think the authors of the VNR Concise Encyclopedia of Mathematics would agree with me that the avtas are a special because in Table V b. on p. 730, the list of composite natural numbers up to 989 with their respective prime factors are exactly the same as the non-prime avtas in the Avta Number Set, and follow the same orderly pattern of multiplication of primes greater than 5 starting with 7² (indicated in Table 1, p. 2 of my monograph).

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 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 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.

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. In Wheel 1: 29 + 2 = 31, the Position 1 avta in Cycle 2. The 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.

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). All higher wheels are subsets of Wheel 1. Thus the Avta Number Set 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 wheels shown below, with 2a white, 4a red or purple, and 6a 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. To classify 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 wheel 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 the ANS circle representation p. 56 with the wheel in one diagram, and the diagram colors have no significance other to the program I wrote which generated them; whereas the colors used in these wheels connote the addition sequence pattern (refer to the corresponding black and white wheel diagram above).

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 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: New York, c 1988].

Geometric Avta Tile Example Arranged by Hand