[seqfan] nth cyclotomic polynomial values modulo n
Peter Lawrence
peterl95124 at sbcglobal.net
Fri Aug 5 07:20:43 CEST 2016
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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