Factoring the Even Trigonometric Polynomials of A269254

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From reference [2] comes a question regarding the linear recurrence of A269254[1], with
; let
be the smallest index
such that
is prime, or
if no such
exists. For what
does the sequence visit
?
Contributed by: Brad Klee (December 2017)
Open content licensed under CC BY-NC-SA
Details
Here, the Chebyshev polynomials are defined in an unusual way [3]:
.
With these conventions, the first few are
and
,
which provides a means to plot all summations as wavefunctions over the domain
, as shown in the plots.
For odd primes, the factorization requires piecewise decomposition. We always have
.
In the exceptional cases where divides
,
,
,
with . Otherwise, if
does not divide
,
,
.
This definition leaves coefficients undetermined. To find the
, next we substitute the ansatz into
,
,
which requires the product-sum identity,
.
On the left- and right-hand sides, we gather constant terms and the coefficients of every . This makes for
total constraints. The system of equations is overdetermined relative to
unknowns
. Reading as
decreases from
, coefficient
first appears as a multiplier of
. It is then trivial to solve all
in succession where
decreases. Yet the solution may not be consistent with all constraints. Define
lower constraints from the constant term and from coefficients of
and
upper constraints from the coefficients of
. The factorization
exists if and only if the solution of the
upper constraints also satisfies the
lower constraints, that is, if the over-determined system of equations is consistent. In practice, we solve the system of equations using matrices and vectors.
With matrices and vectors defined above in code, and with vector containing the
as elements, the upper and lower constraints are written as
,
;
however, when , the system of equations is inconsistent. In these exceptional cases,
.
To explain the case splitting, the Demonstration shows as array plots the dimensional matrix
, the right of the
-dimensional periodic vector
and the
-dimensional periodic vector
. The color rules are:
If , then
and the extra constraints
are always satisfied. If
, then
and the constraint check fails only when
. For all higher cases,
, and the constraint check again fails only when
. In nonexceptional cases, to evaluate the dot product by rows, we first reduce modulo
and then permute the columns by the prime period,
. This technique vastly improves on earlier case-by-case analysis [3, 5, 6].
Solvability of the system of equations is proven by pattern analysis of the matrices here defined and depicted as array plots. Detailed analysis could be the subject for a longer journal-style explication including rigorous proofs of all factorizations presented here.
References
[1] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. "Define a Sequence by , with
; then
Is the Smallest Index
Such That
Is Prime, or
if No Such
Exists." oeis.org/A269254.
[2] H. Havermann, "A269254" (thread). SeqFan. (Oct 21, 2017) list.seqfan.eu/pipermail/seqfan/2017-October/018013.html.
[3] B. Klee, "Proof for A269254" (thread). SeqFan. (Oct 22, 2017) list.seqfan.eu/pipermail/seqfan/2017-October/018016.html.
[4] A. Hone, "A269254, A034807 and Chebyshev Polynomials" (thread). SeqFan. (Oct 27, 2017) list.seqfan.eu/pipermail/seqfan/2017-October/018052.html.
[5] B. Klee, "Re: A269254, A034807 and Chebyshev Polynomials" (thread). SeqFan. (Oct 29, 2017) list.seqfan.eu/pipermail/seqfan/2017-October/018063.html.
[6] B. Klee, "Proof Algorithm for Composite Cases of A269254" from Wolfram Community—A Wolfram Web Resource. (Dec 13, 2017) community.wolfram.com/groups/-/m/t/1209741.
[7] A. Hone, "On a Family of Sequences Related to Chebyshev Polynomials", arXiv:1802.01793[math.nt], 2018. arxiv.org.
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