1232.00 Binomial Symmetry of Scheherazade Numbers

1232.10 Exponential Powers of 1,001: As with all binomials, for example A^{2}+ 2AB +B^{2}, the progressive powers of the 1,001 Scheherazade Number produced by 7 \times 11 \times 13, the product of which, multiplied by itself in successive stages, provides a series of symmetrical reflection numbers. They are not only sublimely rememberable but they resolve themselves into a symmetrical mirror pyramid array:

1. = 1,002,001

2. = 1,003,003,001

1001 4= 1,004,006,004,001

1001 5= 1,005,010,010,005,001

1. = 1,006,015,022,015,006,001

2. = 1,00’7[^63],021,OIOSs’,CQ1,007,001

1001 8 = 1,008,028,056,070,056,028,008,001

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1232.11 The binomial symmetry expands all of its multiples in both left and right directions in reflection balance. Note that the exponential power to which the 1,001 Scheherazade Number is raised becomes the second whole integer from either end. As with (A + B)^{2}= A^{2}+ 2AB +B^{2}, the interior integers consist of expressions and products of the exponent power.

1232.20 Cancellation of “Leftward Spillover”: In the pyramid array of 1,001 Scheherazade Numbers (see Sec. 1232.10), we observe that due to the double-symbol notation of the number 10, the symmetry seems to be altered by the introduction of the leftward accommodation of the two integers of 10 in a single-integer position. For instance,

1001 5= 1,005,010,010,005,001

10 10

= 1,005,000,000,005,001

1 1

= 1,005,000,000,005,001

1232.21 The abacus with its wires and beads taught humans how to fill a column with figures and thereafter to fill additional columns, by convention to the left. The Arabic numerals developed as symbols for the content of the columns. They filled a column and then they emptied it, but the cipher prevented them from using the column for any other notation, and the excessby convention was moved over to the left. This “spillover” can begin earlier or later, depending on the modulus employed. The spillover to the next column begins later when we are employing Modulo 12 than when we are employing Modulo 10. To disembarrass the symmetry of the leftward spillover, the spillover number in the table has been written vertically.

1232.22 The table of the ten successive powers of the 1,001 Scheherazade Number accidentally discloses a series of progressions:

1. in the extreme right-hand column, a progression of zeros;

2. in the fourth column from the right, an arithmetical progression of \frac{N^{2}- N}{2} which we will call triangular; and

3. in the seventh column from the right, a tetrahedral progression.

1232.23 The tetrahedron can be symmetrically or asymmetrically altered to accommodate the four unique planes that produce the fourth-dimensional accommodation of the vector equilibrium. The symmetry disclosed here may very well be four-dimensional symmetry that we have simply expressed in columns in a plane.

1232.24 The number 1,001 looks exciting because we are very close to the binary system of the computers. (We remember that Polynesians only counted to one and two.) The binary yes-no sequence looks so familiar. The Scheherazade Number has all the numbers you have in the binary system. The 1,001-ness keeps persisting throughout the table.

1232.25 The numbers 7 \times 11 \times 13 \times 17 included in the symmetric dividend 510,510 may have an important function in atomic nucleation, since it accommodates all the prime numbers involved in the successive periods.

1232.26 Many mathematicians assume that the integer 1 is not to be counted as a prime. Thus 2, 3, 5, 7, 11, and 13 make a total of six effective primes that may be identified with the fundamental vector edges of the tetrahedron and the six axes of conglomeration of 12 uniradius spheres closest packed around one nuclear sphere, and the fundamental topological abundance of universal lines that always consist of even sets of six.

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Table 1232.21 Cancellation of “Leftward Spillover” to Disclose Basic Reflection Symmetry of Successive Powers of the Scheherazade Numbers: Raising 1001 to ten successive powers, we recognize basic reflective symmetry plus compounding of five basic primes. 7 X 11 X 13 = 1001.

1232.30 Scheherazade Reflection Patterns: