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Research Project: DIVISION ALGORITHM POLYNOMIAL QUOTIENTS. Uneven integers [the dividend [D]] when divided by two [2] produce a quotient [Q] that is invariably an uneven integer provided that the remainder [r] is given by the equation: r = i**{1 + D}. The imaginary " i " and [- i] then become particular zeros of polynomials as seen below. The patterns produced by the zeros of the POLYNOMIAL QUOTIENT EQUATIONS, particularly the complex zeros, may explain the apparently irregular sequence of primes and the complementary irregular sequence of the uneven integers that are composites, that at this time is a perplexing area of mathematics. The prime sequence being: [3, 5, 7, 11, ... ] and the uneven integer composite sequence is: [1, 9, 15, 21, 25, 27, 33, ... ] Noting the absence of two [2], a unique prime in a categorically distinct class and every other even integer, in either sequence. The parameter L[n] provides a zero [root] locus for each zero of a polynomial quotient equation, as L[n] increases from minus infinity through zero to positive values, by integral steps. [including the non-integral value: L[n] = [log[Q[n, N]]/log[2]] - N.] Once the equations have been established, the parameter L[n] may be taken as a continuous variable. This research is based on already very well established mathematical principles. There are no approximations and the equations are exact. The following equations are general, but the instances of these equations are unique to each uneven integer. The equations are derived from the Iterated Division Algorithm with divisor two [2] in every case. The equations in X, [strictly in X**{2}] have the divisor: [X**{2} + 1]. Closely related equations in Y, [strictly in Y**{2}] have the divisor: [Y**{2} - 1]. Statistics: There are asymmetrical polynomial quotients in X and in Y representing Q[n, N]. There are asymmetrical polynomial quotients in X and in Y representing - Q[n, N]. There are symmetrical polynomial quotients in X and in Y representing [Q[n, N] - Q[n, N]. There are anti-symmetrical polynomial quotients in X and in Y representing [Q[n, N] - [ - Q[n, N] - 2.Q[n, N]]. In total, there are eight distinct classes of division algorithm polynomial quotients. The asymmetrical polynomial quotients in X have been tabulated from Q[1, 0] = 1 up to Q[11, 4] = 21. The symmetrical and anti-symmetrical polynomial quotients in X have been tabulated from: Q[1, 0] = 1 [the degenerate case] up to Q[17, 5] = 33 inclusive.

[Multiplication is indicated by a full stop " . " [Am. period].

The division algorithm is: Q[n, N] = 2.Q[N - 1] + r[N - 1], then: Q[N - 1] = 2.Q[N - 2] + r[N - 2] ... counting down. ....................................................                              Q[1] = 2.Q[0] + r[0] ....................................................                      Q[-[L1 - 1]] = 2.Q[- L1] + r[- L1] [ the final iteration for + Q[n, N].] [going to negative subscripts] The square brackets indicate subscripts where the context prescribes this. The uneven integer defined is defined as: Q[n, N] = 2.n - 1 where n = 1, 2, ... satisfies the inequalities: 2**{N} < Q[n, N] < 2**{N + 1} and therefore there is a lower and an upper limit on the integer: n.         There are similar iterations for minus Q[n, N], defined as simply: - Q[n, N]. These iterations run to minus L[2] repeats of the division algorithm after the last proper equation: - Q[1] = 2.[ - Q[0]] + [ - r[0]] ...........................................                   - Q[-[L2 - 1]] = 2.[ - Q[ - L2]] + r[ - L2] [ the final iteration for - Q[n, N].] Noting that Q[N] is identifiable with Q[n, N].

Crucially, the general remainder: r[j - 1] = [i]**{1 + Q[j]} and - r[j - 1] = [i]**{1 - Q[j]} where "i" satisfies: [i]**{2} = - 1. The negative imaginary [ - i] would also be appropriate. The imaginary  "i" is then replaced by the variable: Z, so that: r[j - 1] = Z**{1 + Q[j]} and - r[j] = Z**{1 - Q[j]}. [Putting [a] Q[n, N] = 4.k + 1, [b] Q[n, N] = 4.k + 3 into the above expressions for r[j - 1]] and - r[j - 1] produces remainders that are either - 1 or + 1. This is easily proved.]

In the algebra, the integer: L2 does not survive, but the integer L1 is relabeled L[n] so as to relate it to the particular uneven integer Q[n, N]. In order to explain the derivation, two successive iterations are set down thus: Q[j] = 2.Q[j - 1] + r[j - 1]               Equation. i                 Q[j - 1] = 2.Q[j - 2] + r[j - 2]                Equation. ii Multiplying Equation. ii by two gives: 2.Q[j - 1] = 4.Q[j - 2] + 2.r[j - 2]             Equation. iii Eliminating the term: 2.Q[j - 2] from Equation i and Equation. iii gives: Q[j] = 4.Q[j - 2] + 2.r[j - 2] + r[j - 1]  Equation. iv

This condensation process is started from the final iteration involving L1 until just one equation containing the remainders and Q[ - L1] remains. It may be readily seen that Q[ - L1] is unity. Similarly for the iterations in - Q[n, N]. Really, the improper [in the mathematical sense] iterations are just: 1 = 2.1 - 1 and - 1 = 2.[ - 1] + 1, but have been labelled [Am. labeled].

The [primitive] equations of which there are four in Z, are first presented in the variable: Z and are: Q[n, N] = 2**{N + L[n]} + [2**{N + L[n]} - 2**{N}].Z**{2} + Σ [2**{N - j}.Z**{1 + Q[j]}]                      Equation 1.

Equation 1. can be rearranged to a standard form:

[2**{N + L[n]} - Q[n, N]] + [2**{N + L[n]} - 2**{N}].Z**{2} +

SIGMA [2**{N - j}.Z**{1 + Q[j]}] = 0                           Equation 1.0 Instances of Equation 1.0 are divisible by: [Z**{2} + 1]

The integer: j runs from [1, .. . N] inclusive in Equations 1. and 1.0 and in the following Equations 2., 2.0, 3. and 4.

- Q[n, N] = - 2**{N} + Σ [2**{N - j}.Z**{1 - Q[j]}]          Equation 2.

Taking the term: - Q[n, N] to the right side of Equation 2. and then multiplying through by: Z**{Q[n, N] - 1} to remove negative exponents gives:

[Q[n, N] - 2**{N}].Z**{Q[n, N] - 1} + SIGMA [2**{N - j}.Z**{Q[n, N] - Q[j]}] = 0

Equation 2.0 The polynomial: [Z**{2} + 1] is a divisor of Equation 2.0 The integer: j runs from [1, ... N] inclusive. Note that L[n] is absent from Equation 2. This has been rearranged as Equation 2.0 which can then be immediately solved for a succession of Q[n, N]. Equation 1. is added to Equation 2. and after simplification the following equation results:

[2**{N + L[n]} - 2**{N}].Z**{Q[n, N] + 1}

+ [2**{N + L[n]} - 2**{N}].Z**{Q[n, N] - 1}

+ Σ [2**{N - j}.Z**{Q[n, N] + Q[j]}]

+ Σ [2**{N - j}.Z**{Q[n, N] - Q[j]}] = 0       Equation 3.

The integer: j runs from: [1, ... N] inclusive.

Equation 2. is subtracted from Equation 1. and after simplification, the following equation results: [2**{N + L[n]} + 2**{N} - 2.Q[n, N]].Z**{Q[n, N] - 1}

+ [2**{N + L[n]} - 2**{N}].Z**{Q[n, N] + 1}

+ Σ [2**{N - j}.[Z**{Q[n, N] + Q[j]}]]

- Σ [2**{N - j}.[Z**{Q[n,N] - Q[j]}]] = 0       Equation 4.

The integer: j runs from: [1, ... N] inclusive.

Since Z = X + i.Y, the complex variable, [a] setting Y = 0 makes Z = X. By hypothesis X1 = i and X2 = - i are zeros of the above Equations 3. and 4. and satisfy Equations 1. and 2. so that [X**{2} + 1] = [X - i].[X - [- i]] is a divisor of Equations 3. and 4. Setting X = 0, Z = i.Y or Z**{2} = - Y**{2} and consequently [Y**{2} - 1] is a divisor of equations in Y that are closely related to Equations 1. ... 4. above. Since we can put x = X**{2} and y = Y**{2}, the degree of the equations may be halved. EXAMPLE [a] of a primitive polynomial in X [prior to division by [X**{2} + 1]]: Q[3, 2] = 5, or just Q[2] = 5. This is the least example showing the properties. Here, n = 3, [2.n - 1] = 5, N = 2, Let's take the anti-symmetrical Equation 4. for this example. The integer [j] assumes the values: 1, 2. so we only need to find Q[1]. Rearranging the Division Algorithm Equation above and substituting for the remainder r[N - 1], using [i] rather than Z,  we find that Q[N - 1] = 1/2[Q[N] - [i]**{1 + Q[N]}] Q[1] = 1/2[5 - [i]**{1 + 5}] = 3 Substitution of these values into Equation 4. produces the primitive polynomial for Q[3, 2] = 5 which before simplification is: [2**{2 + L[3]} + 2**{2} - 2.5].Z**{5 - 1} + [2**{2 + L[3]} - 2**{2}].Z**{5 + 1} + 2**{1}.Z**{5 + 3} + 2**{0}.Z**{5 + 5} - 2**{1}.Z**{5 - 3} - 2**{0}.Z**{5 - 5}      = 0

After simplification and rearrangement of terms, this instance of Equation 4. becomes:

Z**{10} + 2.Z**{8} + 4.[2**{L[3]} - 1].Z**{6} + [4.2**{L[3]} - 6].Z**{4} - 2.Z**{2} - 1 = 0

Substitution of Z**{2} = - 1 is found to satisfy the above equation. Algebraic division of this equation by: [Z**{2} + 1] produces the anti-symmetrical quotient equation: Z**{8} + Z**{6} + [4.2**{L[3} - 5].Z**{4} - Z**{2} - 1 = 0 This is seen to agree with the tabulated quotient equation below. END OF EXAMPLE [a] EXAMPLE [b] Using Equation 3. find the primitive polynomial for Q[16, 4] = 31 Note that for convenience of writing, Q[N] is identifiable with Q[n, N]. We have Q[4] = 31 Q[3] = 1/2[Q[4] - i**{1 + Q[4]}] Q[2] = 1/2[Q[3] - i**{1 + Q[3]}] Q[1] = 1/2[Q[2] - i**{1 + Q[2]}] This gives: Q[3] = 15, Q[2] = 7 and Q[1] = 3. Putting these values into Equation 3. after simplification gives:

[2**{L[16] + 4} - 16].Z**{32} + [2**{L[16] + 4} - 16].Z**{30} + 8.Z**{34} + 4.Z**{38} + 2.Z**{46} + Z**{62} + 8.Z**{28} + 4.Z**{24} + 2.Z**{16} + 1 = 0

... an instance of Equation 3. and is divisible by: [Z**{2} + 1] giving the polynomial quotient that is tabulated below for Q[16, 4] = 31. END OF EXAMPLE [b]. In order to find a primitive polynomial from Equations 1.0, 2.0, 3. or 4., given Q[n, N], the values Q[N - 1], ... Q[1] are required and are found from: Q[N - 1] = 1/2[Q[n, N] - i**{1 + Q[n, N]}], starting with Q[n, N], then Q[N - 2] is found by replacing Q[n, N] by Q[N - 1] in the above formula. Etc. Apparently, the parameter L[n] can have any value, including integral values, from minus infinity through zero up to some finite positive value. In the polynomial quotient equations below, a particular value value of L[n] is found to nullify the center [Am. center] term. This being: L[n] = [log[Q[n, N]]/log[2]] - N Instances of the above primitive Equations 3. and 4. in X have been divided by [X**{2} + 1] and tabulated in columnar form below: Each quotient equation is implicitly equated to zero.

ASYMMETRICAL POLYNOMIAL QUOTIENTS IN X Asymmetrical equations are tabulated immediately below: Asymmetrical [from Equation 2.0]    Asymmetrical [from Equation 1.0] Q[1, 0] = 1 Exp X        Coefficient X        Exp X             Coefficient X    0             1                      0       [2**{L[1] - 1] Q[2, 1] = 3 Exp X      Coefficient X          Exp X           Coefficient X    0             1                      2                 1 0      [2**{L[2] + 1} -3] Q[3, 2] = 5 Exp X      Coefficient X          Exp X           Coefficient X    2             1                      4                 1 0            1                      2                 1                                         0       [2**{L[3] + 2} - 5]

Q[4, 2] = 7 Exp X      Coefficient X          Exp X           Coefficient X    4             3                      6                 1 2          - 1                      4               - 1    0             1                      2                 3                                         0       [2**{L[4] + 2} - 7] Q[5, 3] = 9 Exp X    Coefficient X            Exp X           Coefficient X    6             1                      8                 1 4            3                      6               - 1    2           - 1                      4                 3    0             1                      2                 1                                         0       [2*{L[5] + 3} - 9]

Q[6, 3] = 11 Exp X       Coefficient X           Exp X         Coefficient X    8            3                       10                1 6           1                        8              - 1    4            1                        6                1    2          - 1                        4                1    0            1                        2                3                                          0    [2**{L[6] + 3} - 11]

Q[7, 3] = 13 Exp X    Coefficient X              Exp X         Coefficient X    10          5                        12                1 8       - 1                        10              - 1     6          1                         8                1     4          1                         6                1     2        - 1                         4              - 1     0          1                         2                5                                          0    [2**{L[7] + 3} - 13]

Q[8, 3] = 15 Exp X   Coefficient X              Exp X         Coefficient X    12          7                       14               1 10       - 3                       12             - 1     8          3                       10               1     6        - 1                        8             - 1     4          1                        6               3     2        - 1                        4             - 3     0          1                        2               7                                         0     [2***{L[8] + 3} - 15]

Q[9, 4] = 17 Exp X   Coefficient X              Exp X         Coefficient X    14          1                       16                1 12         7                       14              - 1    10        - 3                       12                1     8          3                       10              - 1     6        - 1                        8                3     4          1                        6              - 3     2        - 1                        4                7     0          1                        2                1                                         0    [2**{L[9] + 4} - 17]

Q[10, 4] = 19 Exp X  Coefficient X               Exp X         Coefficient X    16           3                      18                1 14          5                      16              - 1    12         - 1                      14                1    10           1                      12              - 1     8           1                      10                1     6         - 1                       8                1     4           1                       6              - 1     2         - 1                       4                5     0           1                       2                3                                         0    [2**{L[10] + 4} - 19]

Q[11, 4] = 21 Exp X   Coefficient X               Exp X         Coefficient X   18           5                       20                1 16          3                       18              - 1   14           1                       16                1   12         - 1                       14              - 1   10           1                       12                1    8           1                       10                1    6         - 1                        8              - 1    4           1                        6                1    2         - 1                        4                3    0           1                        2                5                                         0    [2**{L[11] + 4} - 21] to infinity. ................................................................. ASYMMETRICAL POLYNOMIAL QUOTIENTS IN Y  Asymmetrical [from Equation 2.0] Asymmetrical [from Equation 1.0] Q[1, 0] = 1 Exp Y   Coefficient Y          Exp Y           Coefficient Y   0               1               0           [2**{L[1]} - 1]

Q[2, 1] = 3 Exp Y      Coefficient Y       Exp Y           Coefficient Y   0               1               2               1 0        - [2**{L[2] + 1} - 3]

Q[3, 2] = 5 Exp Y       Coefficient Y      Exp Y        Coefficient Y   2               1               4               1 0            - 1               2             - 1                                   0       [2**{L[3] + 2} - 5]

Q[4, 2] = 7 Exp Y        Coefficient       Exp Y       Coefficient Y   4                3              6               1 2               1              4               1   0                1              2               3                                   0        [2**{L[4] + 2} - 7] Q[5, 3] = 9 Exp Y        Coefficient Y     Exp Y       Coefficient Y   6                1              8               1 4             - 3              6               1   2              - 1              4               3   0              - 1              2             - 1                                   0        [2**{L[5] + 3} - 9]

Q[6, 3] = 11 Exp Y    Coefficient Y         Exp Y       Coefficient Y   8                3              10              1 6             - 1               8              1   4                1               6              1   2                1               4            - 1   0                1               2              3                                    0      [2**{L[6] + 3} - 11] to infinity. ............................................................. SYMMETRICAL AND ANTI-SYMMETRICAL POLYNOMIAL QUOTIENTS IN X.   Q[1, 0] = 1 Exp X         Coefficient X    0             [2**{L[1]} - 1]      [the degenerate case]

Symmetrical [from Equation. 3]                    Anti-symmetrical  [from Eq. 4.]  Q[2, 1] = 3 Exp X         Coefficient X                  Exp X       Coefficient X    4               1                             4            1 2   [2**{L[2] + 1} - 3]                      2 [2**{L[2] + 1} - 3] 0              1                             0          - 1

Q[3, 2] = 5 Exp X           Coefficient X               Exp X         Coefficient X   8                 1                            8              1 6                1                            6              1   4               [2**{L[3] + 2} - 5]            4   [2**{L[3] + 2} - 5] 2                1                            2            - 1   0                 1                            0            - 1

Q[4, 2] = 7 Exp X           Coefficient X               Exp X          Coefficient X    12               1                          12                 1 10            - 1                          10               - 1     8               3                           8                 3     6             [2**{L[4] + 2} - 7]           6      [2**{L[4] + 2] - 7] 4              3                           4               - 3     2             - 1                           2                 1     0               1                           0               - 1

Q[5, 3] = 9 Exp X           Coefficient X               Exp X        Coefficient X    16                1                          16                 1 14             - 1                          14               - 1    12                3                          12                 3    10                1                          10                 1     8             [2**{L[5] + 3} - 9]            8       [2**{L[5] + 3] - 9] 6               1                           6                - 1     4                3                           4                - 3     2              - 1                           2                  1     0                1                           0                - 1

Q[6, 3] = 11 Exp X     Coefficient X                     Exp X         Coefficient X      20            1                             20                1 18         - 1                             18              - 1       16            1                             16                1      14            1                             14                1      12            3                             12                3      10     [2**{L[6] + 3} - 11]                 10     [2**{L[6] + 3} - 11] 8           3                              8              - 3       6            1                              6              - 1       4            1                              4              - 1       2          - 1                              2                1       0            1                              0              - 1

Q[7, 3] = 13 Exp X        Coefficient X                 Exp X           Coefficient X      24             1                            24               1 22          - 1                            22             - 1      20             1                            20               1      18             1                            18               1      16           - 1                            16             - 1      14             5                            14               5      12  [2**{L[7] + 3} - 13]                    12    [2**{L[7] + 3} - 13] 10            5                            10             - 5       8           - 1                             8               1       6             1                             6             - 1       4             1                             4             - 1       2           - 1                             2               1       0             1                             0             - 1

Q[8, 3] = 15 Exp X       Coefficient X                     Exp X         Coefficient X       28                 1                           28               1 26              - 1                           26             - 1       24                 1                           24               1       22               - 1                           22             - 1       20                 3                           20               3       18               - 3                           18             - 3       16                 7                           16               7       14          [2**{L[8] + 3} - 15]               14         [2**{L[8] + 3} - 15] 12                7                           12             - 7       10               - 3                           10               3        8                 3                            8             - 3        6               - 1                            6               1        4                 1                            4             - 1        2               - 1                            4               1        0                 1                            0             - 1      Q[9, 4] = 17 Exp X            Coefficient X                Exp X         Coefficient X       32                  1                          32               1 30               - 1                          30             - 1       28                  1                          28               1       26                - 1                          26             - 1              24                  3                          24               3       22                - 3                          22             - 3       20                  7                          20               7       18                  1                          18               1       16             [2**{L[9] + 4}  -  17]          16    [2**{L[9] + 4} -  17] 14                 1                          14             - 1       12                  7                          12             - 7       10                - 3                          10               3        8                  3                           8             - 3        6                - 1                           6               1        4                  1                           6             - 1        2                - 1                           2               1        0                  1                           0             - 1     Q[10, 4] = 19 Exp X              Coefficient X               Exp X      Coefficient X      36                    1                         36                1 34                 - 1                         34              - 1      32                    1                         32                1      30                  - 1                         30              - 1      28                    1                         28                1      26                    1                         26                1      24                  - 1                         24              - 1      22                    5                         22                5        20                    3                         20                3      18              [2**{L[10] + 4} - 19]           18     [2**{L[10] + 4} - 19] 16                   3                         16              - 3       14                    5                         14              - 5      12                  - 1                         12                1      10                    1                         10              - 1                       8                    1                          8              - 1       6                  - 1                          6                1       4                    1                          4              - 1       2                  - 1                          2                1       0                    1                          0              - 1

Q[11, 4] = 21 Exp X             Coefficient X          Exp X       Coefficient X                           40                  1                      40                1 38               - 1                      38              - 1     36                  1                      36                1     34                - 1                      34              - 1     32                  1                      32                1     30                  1                      30                1     28                - 1                      28              - 1     26                  1                      26                1     24                  3                      24                3     22                  5                      22                5     20             [2**{L[11] + 4} - 21]       20  [2**{L[11] + 4} - 21] 18                 5                      18              - 5     16                  3                      16              - 3     14                  1                      14              - 1     12                - 1                      12                1     10                  1                      10              - 1      8                  1                       8              - 1      6                - 1                       6                1      4                  1                       4              - 1      2                - 1                       2                1      0                  1                       0              - 1  Q[12, 4] = 23 Exp X       Coefficient X   Exp X  Coefficient X                                              44                1           44       1 42             - 1           42     - 1   40                1           40       1                             38              - 1           38     - 1   36                1           36       1   34              - 1           34     - 1   32                3           32       3   30              - 3           30     - 3   28                3           28       3   26                1           26       1   24                7                 24            7   22        [2**{L[12] + 4} - 23]     22    [2**{L[12] + 4} - 23] 20               7                 20          - 7   18                1                 18          - 1   16                3                 16          - 3   14              - 3                 14            3   12                3                 12          - 3   10              - 1                 10            1    8                1                  8          - 1    6              - 1                  6            1    4                1                  4          - 1    2              - 1                  2            1    0                1                  0          - 1

Q[13, 4] = 25 Exp X         Coefficient X      Exp X       Coefficient X    48               1                48            1 46            - 1                46          - 1    44               1                44            1                  42             - 1                42          - 1    40               1                40            1    38             - 1                38          - 1    36               3                36            3    34             - 3                34          - 3    32               3                32            3    30               1                30            1    28             - 1                28          - 1    26               9                26            9    24      [2**{L[13] + 4] - 25]     24   [2**{L[13] + 4} - 25] 22              9                22          - 9    20             - 1                20            1    18               1                18          - 1    16               3                16          - 3    14             - 3                14            3    12               3                12          - 3    10             - 1                10            1     8               1                 8          - 1     6             - 1                 6            1     4               1                 4          - 1     2             - 1                 2            1     0               1                 0          - 1

Q[14, 4] = 27 Exp X       Coefficient X    Exp X          Coefficient X   52               1            52              1 50            - 1            50            - 1   48               1            48              1   46             - 1            46            - 1   44               1            44              1   42             - 1            42            - 1   40               1            40              1   38               1            38              1   36             - 1            36            - 1   34               1            34              1   32               3            32              3   30             - 3            30            - 3   28              11            28             11   26   [2**{L[14] + 4} - 27]    26   [2**{L[14] + 4} - 27] 24             11            24           - 11   22             - 3            22              3   20               3            20            - 3   18               1            18            - 1   16             - 1            16              1   14               1            14            - 1   12               1            12            - 1   10             - 1            10              1    8               1             8            - 1    6             - 1             6              1    4               1             4            - 1    2             - 1             2              1    0               1             0            - 1

Q[15, 4] = 29 Exp X    Coefficient X      Exp X      Coefficient X    56             1             56               1 54          - 1             54             - 1    52             1             52               1    50           - 1             50             - 1    48             1             48               1    46           - 1             46             - 1    44             1             44               1    42             1             42               1    40           - 1             40             - 1    38             1             38               1    36           - 1             36             - 1    34             5             34               5    32           - 5             32             - 5    30            13             30              13    28  [2**{L[15] + 4} - 29]    28   [2**{L[15] + 4} - 29] 26           13             26            - 13    24           - 5             24               5    22             5             22             - 5    20           - 1             20               1    18             1             18             - 1    16           - 1             16               1    14             1             14             - 1    12             1             12             - 1    10           - 1             10               1     8             1              8             - 1     6           - 1              6               1     4             1              4             - 1     2           - 1              2               1     0             1              0             - 1

Q[16, 4] = 31 Exp X      Coefficient X      Exp X          Coefficient X   60                1            60               1 58             - 1            58             - 1   56                1            56               1   54              - 1            54             - 1   52                1            52               1   50              - 1            50             - 1   48                1            48               1   46              - 1            46             - 1   44                3            44               3   42              - 3            42             - 3   40                3            40               3   38              - 3            38             - 3   36                7            36               7   34              - 7            34             - 7   32               15            32              15   30   [2**{L[16] + 4} - 31]     30   [2**{L[16] + 4} - 31] 28              15            28            - 15   26              - 7            26               7   24                7            24             - 7   22              - 3            24               3   20                3            20             - 3   18              - 3            18               3   16                3            16             - 3   14              - 1            14               1   12                1            12             - 1   10              - 1            10               1    8                1             8             - 1    6              - 1             6               1    4                1             4             - 1    2              - 1             2               1    0                1             0             - 1  Q[17, 5] = 33 Exp X      Coefficient     X Exp X          Coefficient X   64                1           64                1 62             - 1           62              - 1   60                1           60                1   58              - 1           58              - 1   56                1           56                1   54              - 1           54              - 1   52                1           52                1   50              - 1           50              - 1   48                3           48                3   46              - 3           46              - 3   44                3           44                3   42              - 3           42              - 3   40                7           40                7   38              - 7           38              - 7   36               15           36               15   34                1           34                1   32    [2**{L[16] + 5} - 33]   32    [2**{L[16] + 5} - 33] 30               1           30              - 1   28               15           28             - 15   26              - 7           26                7   24                7           24              - 7   22              - 3           22                3   20                3           20              - 3   18              - 3           18                3   16                3           16              - 3   14              - 1           14                1   12                1           12              - 1   10              - 1           10                1    8                1            8              - 1    6              - 1            6                1    4                1            4              - 1    2              - 1            2                1    0                1            0              - 1

to infinity. ............................................................... SYMMETRICAL AND ANTI-SYMMETRICAL POLYNOMIAL QUOTIENTS IN Y.  Polynomial Quotient Equations in Y  [implicitly equated to zero]

Symmetrical                        Anti-symmetrical Q[2, 1] = 3 Exp Y       Coefficient Y           Exp Y        Coefficient Y    4             1                      4             1 2       - [2**{L[2] + 1} - 3]       2      - [2**{L[2] + 1} - 3] 0            1                      0           - 1   Q[3, 2] = 5 Exp Y       Coefficient Y           Exp Y       Coefficient Y    8              1                     8             1 6           - 1                     6           - 1    4          [2**{L[3] + 2} - 5]       4      [2**{L[3] + 2} - 5] 2           - 1                     2             1    0              1                     0           - 1

Q[4, 2] = 7 Exp Y        Coefficient Y         Exp Y       Coefficient Y    12              1                    12            1 10             1                    10            1     8              3                     8            3     6      - [2**{L[4] + 2} - 7]         6     - [2**{L[4] + 2} - 7] 4             3                     4          - 3     2              1                     2          - 1     0              1                     0          - 1

Q[5, 3] = 9 Exp Y       Coefficient Y         Exp Y     Coefficient Y     16             1                    16            1 14            1                    14            1     12             3                    12            3     10           - 1                    10          - 1      8    [2**{L[5] + 3} - 9]            8     [2**{L[5] + 3} - 9] 6          - 1                     6            1      4             3                     4          - 3      2             1                     2          - 1      0             1                     0          - 1    Q[6, 3] = 11 Exp Y       Coefficient Y        Exp Y       Coefficient Y     20             1                    20            1 18            1                    18            1     16             1                    16            1     14           - 1                    14          - 1     12             3                    12            3     10   - [2**{L[6] + 3} - 11]         10     - [2**{[6] + 3} - 11] 8            3                     8          - 3      6           - 1                     6            1      4             1                     4          - 1      2             1                     2          - 1      0             1                     0          - 1

Q[7, 3] = 13 Exp Y       Coefficient Y        Exp Y       Coefficient Y      24             1                    24            1 22            1                    22            1      20             1                    20            1      18           - 1                    18          - 1      16           - 1                    16          - 1      14           - 5                    14          - 5      12    [2**{L[7] + 3} - 13]          12    [2**{L[7] + 3} - 13] 10          - 5                    10            5       8           - 1                     8            1       6           - 1                     6            1       4             1                     4          - 1       2             1                     2          - 1       0             1                     0          - 1  to infinity. ..............................................................

The anti-symmetrical polynomial quotient equation can be inferred from the symmetrical polynomial quotient equation coefficients. Only the coefficients of the symmetrical polynomial quotient equation from the term involving X**{2.Q[n, N] - 2} down to the term involving X**{Q[n, N] - 1} are required to construct the polynomials. The remaining coefficients are all self-evident. Similarly with the symmetrical and anti-symmetrical equations in Y. The symmetrical and anti-symmetrical quotient equations, taken from the original paper, have been tabulated in all cases up to Q[17, 5] = 33 in X and up to Q[7, 3] = 13 in Y.     The matter of interest here would be to find the amplitudes [moduli/moduli] and arguments of the complex zeros and any real zeros of the above Polynomial Quotient Equations that uniquely represent uneven integers, for various values of L[n] such as L[n] = 0, minus infinity, [log[Q[n, N]]/log[2]] - N etc. The zeros labelled according to the Q[n, N] so represented [Am. labeled] could be set on an Argand diagram and analyzed. [This is not an exact copy of the shorter version done for Wikipedia under a different user name] SHAWWPG19410425 01:47, 29 January 2011 (U.T.C.). The work above only needs to be reworked by others and the same Equations 1., ...4 found. This would constitute verification. The above results are simply the outcome of the application of well established elementary maths that could have been done previously.

The solutions all form a regular pattern of complex numbers.

SHAWWPG19410425 12:22, 29 January 2011 (U.T.C.)SHAWWPG19410425 09:26, 30 January 2011 (U.T.C.) SHAWWPG19410425 11:44, 30 January 2011 (U.T.C.)SHAWWPG19410425 17:22, 31 January 2011 (U.T.C.)SHAWWPG19410425 18:38, 31 January 2011 (U.T.C.) SHAWWPG19410425 18:27, 1 February 2011 (U.T.C.) SHAWWPG19410425 15:06, 6 February 2011 (U.T.C.) SHAWWPG19410425 18:16, 6 February 2011 (U.T.C.)SHAWWPG19410425 21:52, 6 February 2011 (U.T.C.)SHAWWPG19410425 18:04, 14 February 2011 (U.T.C.)SHAWWPG19410425 21:28, 27 March 2011 (U.T.C.)SHAWWPG19410425 21:29, 29 March 2011 (U.T.C.)SHAWWPG19410425 15:43, 30 March 2011 (U.T.C.)SHAWWPG19410425 13:36, 31 March 2011 (U.T.C.)SHAWWPG19410425 12:02, 5 April 2011 (U.T.C.)SHAWWPG19410425 (discuss • contribs) 19:37, 2 February 2014 (UTC)