# [seqfan] Re: SeqFan Digest, Vol 95, Issue 1

Peter Lawrence peterl95124 at sbcglobal.net
Sun Sep 4 17:48:52 CEST 2016

```Jan Orwat,
Many thanks,

as a software professional by day (and a mathematician only by night)
I can't believe I made such a rookie C programming mistake,
the C language "%" remainder operator is broken for negative dividends
and does not compute modulo arithmetic, after fixing this I get the
same numbers as you. The patterns in these numbers are much simpler.
Again, many thanks.

Peter Lawrence.

>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.
>>
>>
>>
>> --

```