Research Wikiversity: DIVISION ALGORITHM POLYNOMIALQUOTIENTS
The mathematical topic: DIVISION ALGORITHM POLYNOMIAL QUOTIENTS already exists and has been comprehensively typed out on a Talk Page. Resulting symmetrical and antisymmetrical polynomial quotients have been tabulated from the integer unity [1], the degenerate case, then for the series of Q's : [3, 5, 7, 9, ... 33]. The polynomials continue to any uneven integer however large but always finite.
The derivation is rather lengthy but the principle is that any uneven integer: Q[N] = 2.n + 1, identically, can be divided by two [2] in such a way that the remainder depends only on Q[N] and is either + 1 or - 1. The integer: [N] is such that: 2**{N} < Q[N] < 2**{N + 1} . The two inequalities set limits on the integer: [n] where [n] takes values: [0, 1, 2, ...]. This division by two invariably yields a quotient that is also an uneven integer.
The two extreme values of the integer [n] that satisfy the above inequalities are: n[minimum] = 2**{N - 1} and n[maximum] = 2**{N} - 1, for N equal or greatr than unity..
Iteration of the Division Algorithm as above, starting with Q[N] = 2.Q[N - 1] + r[N - 1] down to iterations of the form: 1 = 2.1 - 1 and similar for - Q[N] with proper counting of the iterations yields polynomials. The integer: [N] counts down.
The general remainder: r[j - 1] = [i]**{1 + Q[j]}. " i " satisfies: i^2 = - 1, using the more conventional " ^ " here rather than " ** " to represent exponentiation. This is generalized to:
r[j - 1] = Z**{1 + Q[j]}
Negative Q[j] given by - Q[j] gives the remainder:
- r[j - 1] = [i]**{1 - Q[j]} This is generalized to:
- r[j - 1] = Z**{1 - Q[j]}
Irrespective of whether Q[N] or - Q[N] is divided by two iteratively, the divisor is always plus two [ + 2].
The iteration: 1 = 2.1 - 1 is repeated integer: L times and this produces the constant: L[n], that can have any fixed value in the final polynomial equations. There is another integer L[2] involved in the iterations of - Q[N], but L[2] disappears in the algebra.
The iterations are algebraically " condensed " so eliminating every Q[j] with the exception of Q[N, n] and the exception of the final Q[j] of value unity. For each value of Q[N, n], there are two primitive asymmetrical polynomials which when combined either by addition or by subtraction produce the primitive polynomials below. The primitive equation leading to symmetrical quotient polynomials is:
[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
and the primitive equation leading to antisymmetrical quotient polynomials is:
[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
The integer: j in the summations runs j = 1, ... N inclusive. Q[N, n] is identifiable with Q[N].
For particular values of Q[N, n] that are consistent with the integer N [see the above lower and upper limits on the integer [n]], the above two primitive equations are easily divisible by the divisor: [Z**{2} + 1], with remainder zero. In either case, a quotient polynomial equation results.
Unlike the primitive polynomials, there does not appear to be a general form of the quotient polynomial for a non-specific Q[N, n].
The next stage would be to solve the tabulated quotient polynomials taking values of Q: [3, 5, ... 33] using a suitable polynomial solver computer program, for simple values of L[n] such as L[n] = 0, noting that the substitution: z = Z**{2} halves the degree of the quotient polynomial in every case.
A plot of the labeled zeros of the quotient polynomial equations may produce a further insight into prime and non-prime uneven integers that have two or more prime factors.
[TO BE CONTINUED] SHAWWPG19410425 11:20, 9 June 2011 (UTC)SHAWWPG19410425 21:18, 24 July 2011 (UTC)SHAWWPG19410425 (talk) 20:55, 15 June 2012 (UTC)