[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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