Factoring the Even Trigonometric Polynomials of A269254

show

x

k

y

k

factorization over 

i

wavefunction decomposition

V

1

|

′

M

|

V

u

: constraint check

M

u

-1

M

u

V

1

check

all k×K elements

reduce modulo period 2k+1

permute columns modulo p

prime p

2

3

5

7

11

13

17

19

23

29

index k

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1 +



2

+



4

+



6

=

(1 +



1

+



2

+



3

) × (-1 +



1

-



2

+



3

)

From reference [2] comes a question regarding the linear recurrence of A269254 [1],

s

k

=n

s

k-1

-

s

k-2

with

s

0

=1,

s

1

=n+1

; let

a(n)

be the smallest index

k

such that

s(k)

is prime, or

-1

if no such

k

exists. For what

n

does the sequence visit

-1

?

Andrew Hone noticed that

a(n)=-1

seems only to occur when

n=



p

(j)

for prime

p

,

j>2

and



p

(j)

is a trigonometric polynomial of order

p

[3] (see the details for more definitions). In these cases, we have proven that the solution

z

k

of the recurrence

z

k

=



p

(j)

z

k-1

-

z

k-2

with

z

0

=1,

z

1

=1+



p

(j)

can be written as a simple summation,

z

k

=1+

k

∑

i=1



pi

(j)

. This Demonstration gives an algorithm that uses linear algebra to calculate the polynomial factorization of

s

k

=

z

k

in all cases

n=



p

(j)

for prime

p

and

j>2

. This Demonstration helps to extend earlier empirical observations and proofs for cases

p=2,3

[4, 5, 6].