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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)

title: User:SHAWWPG19410425


text:
Automata theory

Sometimes also referred to as the Theory of Computation or a superset thereof, Automata Theory is a field within Set Theory and Computer Science, and serves as the theoretical framework through which computers and modern computing came about. Although somewhat esoteric, as indicated by the strange names you will encounter throughout, it has many applications, ranging across programming machines, string computations (such as regular expressions and parsing), and problem solving of decision problems. It can also be an entertaining exercise in programming to try and solve common problems (such as number addition) by writing machines to solve them. The machines constructed will largely resemble flow charts and will behave very similarly.

Before starting this course, it is recommended you have some familiarity with Set Theory and Set Notation, although the first few uses of it will briefly review the meanings of the notations. You should also be somewhat familiar with Formal Definitions and Formal Proofs, or at least understand what they are. Even though this is a topic within Computer Science, an understanding of computers, programming, or the likes is not necessary for this. Many topics of this depth use a lot of jargon, formulas, and symbols, and can get very confusing very fast for someone who doesn't have a background in that stuff. I try my best to avoid that and keep it as humanly readable as possible. If there is something you don't understand, feel free to ask about it on the Discuss page or direct it to the instructor.

Terminology and symbols
Throughout the different books and materials on Automata Theory and computability, different symbols and terminology may be used, as there is no standard for symbols and terminology. Generally, however, the concepts are the same, there's just slight deviations in symbols or terminology, so it's not too difficult to decipher. Herein we will use one set of symbols and terminology, and leave it as an exercise to the reader to decipher other symbols and terminology used in other sources when they are encountered.

Introduction
First we must define the basic components that will make up our definitions.
 * Alphabet - A finite set of symbols or characters. Denoted uppercase Greek letter Sigma: ∑
 * String - A finite sequence of symbols from some alphabet ∑, usually surrounded by "quotes". If your alphabet is { a, b, c }, a possible string would be "acbab".
 * The empty string, or a string with no characters in it, is denoted with lowercase Greek letter lunate epsilon: ϵ = ""
 * The set of all possible strings over an alphabet ∑ is denoted ∑* (the Kleene star)
 * For an alphabet ∑ = { a, b }, ∑* would be { ϵ, "a", "b", "aa", "ab", "ba", "bb", "aaa", ... } (the set is infinite, but only consists of a's and b's)
 * Language - A set of valid strings over a finite alphabet. Denoted L, and with the context of alphabet ∑ it can be denoted ∑L.
 * Empty Language - A language containing no strings, denoted {} or ∅.
 * Do not confuse with the empty string, which is not in the empty language. ϵ ≠ ∅ and ϵ ∉ ∅.
 * Natural Languages (like the English language) will not be dealt with. They are far too complex and oftentimes can't even be expressed with a formal grammar.

Latin alphabet letters frequently used in notation:
 * c - usually represents symbols or characters.
 * s, t, w - usually represent strings
 * x, y - usually represent variables
 * i, j, k, n - usually represents numbers

Function notation:
 * |s| - Length function. Returns the number of symbols in s. For instance, |"hello world"| = 11
 * #c(s) - Occurrence function. Returns the number of times c occurs in s. For instance, #l("hello world") = 3
 * s||t or st - Concatenation function. Returns a single string whose prefix is s and whose suffix is t, essentially combining s and t to form one string.
 * Concatenation is associative. (st)w = s(tw).
 * ϵ is identity for concatenation. ϵs = sϵ = s. That is, any string s concatenated with ϵ remains s.
 * si - Replication function. Returns the string s repeated i times. For instance, a3 = aaa and "ab"3 = "ababab".
 * w0 = ϵ
 * wi+1 = wiw
 * sR - Reversal function. "abc"R = "cba"
 * Reversing any string of length 1 or 0 will return the original string (in the latter case, ϵ).
 * (st)R = tRsR

String terminology:
 * Substring - A sequence of consecutive symbols from a string. "beg" is a substring of "begin".
 * A substring t is a proper substring of s iff s ≠ t. That is, "hi" is a substring of "hi", but not a proper substring.
 * Prefix - A substring that comes at the beginning of a string. Formally, s is a prefix of t iff ∃w ∈ Σ*(t = sw)
 * Suffix - A substring that comes at the end of the string. Formally, s is a suffix of t iff ∃w ∈ Σ*(t = ws)
 * Proper - A proper substring is any substring s of t where s ≠ t. Likewise, you can also have a proper prefix and a proper suffix.
 * ϵ is a substring, prefix, and suffix of every string.

Next
Next we will discuss Finite Automata.

Other topics that will be discussed for automata theory:
 * Finite Automata
 * Deterministic and Non-deterministic (not to be confused with P vs. NP)
 * Regular Expressions
 * Regular Grammars
 * Regular Languages
 * Deciding whether a language is Regular
 * Context-Free Grammars
 * Deciding whether a language is Context-Free
 * Push-down Automata
 * Turing Machines
 * Decidability (Halting, etc)
 * Proofs

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Automata theory


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User talk:Kempoka8h

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 04:14, 29 January 2011 (UTC)

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