[seqfan] Re: nth cyclotomic polynomial values modulo n

[Jan Orwat]

Dear Peter,

For C30(x) I've got a row of all ones (mod 30).
Your result C30(1) -> 17, suggests you use other coefficients than [ 1, 1,
0,-1,-1,-1, 0, 1, 1].
Check if C30(x) equals C15(-x).

Best regards,
Jan Orwat

2016-08-05 7:20 GMT+02:00 Peter Lawrence <peterl95124 at sbcglobal.net>:

>
> I was playing around with cyclotomic polynomials,
> in particular I was wondering how to verify my calculations
> of their coefficients without using floating-point arithmetic
> to evaluate their supposed roots
>
> and wondered about the values of Cn(x) modulo n
> evaluated for x in 0..n-1,
>
> I did not seem to find these values in OEIS,
> did I compute them incorrectly ?
>
> there are some obvious patterns in the numbers I computed with modulo n
> arithmetic
> Cp(x) ---> 1,0,1,1,1,1,.....
> Cp^e(x) :  all 1's except Cn(1), Cn(1+p), Cn(1+2p), ..., Cn(1+p^e-p) ---> p
> Cn(x) with n = 2q with q odd:  Cn(q-1), Cn(2q-1) ---> q
>
> but things seem to get wild around C30(x),
>
> would anyone else like to verify the triangle of values I came up with
> for n = 1,..., 30  ?
>    1
>    1  0
>    1  0  1
>    1  2  1  2
>    1  0  1  1  1
>    1  1  3  1  1  3
>    1  0  1  1  1  1  1
>    1  2  1  2  1  2  1  2
>    1  3  1  1  3  1  1  3  1
>    1  1  1  1  5  1  1  1  1  5
>    1  0  1  1  1  1  1  1  1  1  1
>    1  1  1  1  1  1  1  1  1  1  1  1
>    1  0  1  1  1  1  1  1  1  1  1  1  1
>    1  1  1  1  1  1  7  1  1  1  1  1  1  7
>    1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
>    1  2  1  2  1  2  1  2  1  2  1  2  1  2  1  2
>    1  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
>    1  1  3  1  1  3  1  1  3  1  1  3  1  1  3  1  1  3
>    1  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
>    1  1  5  5  1  1  1  5  5  1  1  1  5  5  1  1  1  5  5  1
>    1  1  7  1  7  1  1  1  1  7  1  7  1  1  1  1  7  1  7  1  1
>    1  1  1  1  1  1  1  1  1  1 11  1  1  1  1  1  1  1  1  1  1 11
>    1  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
>    1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
>    1  5  1  1  1  1  5  1  1  1  1  5  1  1  1  1  5  1  1  1  1  5  1  1
> 1
>    1  1  1  1  1  1  1  1  1  1  1  1 13  1  1  1  1  1  1  1  1  1  1  1
> 1 13
>    1  3  1  1  3  1  1  3  1  1  3  1  1  3  1  1  3  1  1  3  1  1  3  1
> 1  3  1
>    1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
> 1  1  1  1
>    1  0  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
> 1  1  1  1  1
>    1 17  1  1  1 21  1  1  1 25  1 27  1  1 15  1  1  1  1  5 21  1  1  1
> 25  1  1  1  1 15
>
> if these values are correct I'll go ahead and submit the sequence,
> then see if I can prove the observations,
> but the last line above for 30 seems without pattern,
>
>
> thanks,
> Peter Lawrence.
>
>
>
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