20170407, 00:35  #1 
"Sam"
Nov 2016
326_{10} Posts 
Polynomial whose coefficients add up to n defining Cyclotomic field K.
Let n be an integer defining the cyclotomic properties of K (meaning that n is a factor of the cyclotomic polynomial C_K(x) evaluated at some x value). How many polynomials P(x), the same degree as C_K(x) their coefficients add up to n? For instance, choosing the cyclotomic field 3, C_3(x) = x^2+x+1, and x = 7, 19 is a factor of 7^2+7+1. How many polynomials P(x) of the form ax^2+bx+c defining the same field as x^2+x+1 is it the case that a+b+c = 19 where a, b, c are integers 19 <= (a, b, c) <= 19? Thanks for help, comments, and clarification.
Last fiddled with by carpetpool on 20170407 at 00:35 
20170407, 00:50  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
d,d,19 20*6 polynomials with ordering changes like this ( edited to include +0 and 0) d,(d1),18 where d is positive, .... etc. edit2: turns out there are 400 possibilities to look through ( as some have only one order that is unique better than searching all 59319 {a,b,c} in that range by hand though. edit 2 + : Code:
my(a=[19..19]);b=setbinop((x,y)>concat(x,y),a);b=setbinop((x,y)>concat(x,y),b,a);b=select(r>vecsum(r)==19,b) Last fiddled with by science_man_88 on 20170407 at 01:22 

20170407, 02:09  #3 
"Sam"
Nov 2016
2×163 Posts 
So just looking on the conditions mod 3 we have:
a = 1 b = 2 c = 1 a = 0 b = 0 c = 1 I also thought about another exception: n may also define properties of the cyclotomic field K if and only if each prime power p^k dividing n is either 0 or 1 mod n. So in this sense, factors such as 2^(2*n), 5^(2*n), 11^(2*n), 17^(2*n), 23^(2*n), 29^(2*n)..., etc. would be allowed, but I don't know weather this would make problem harder. Depending on (prime) K, we also allow n to divide: for K = 5, 2^(4*n), 3^(4*n), 7^(4*n), 13^(4*n), 17^(4*n), 19^(2*n), 23^(4*n), 29^(2*n)..., etc. for K = 7, 2^(3*n), 3^(6*n), 5^(6*n), 11^(3*n), 13^(2*n), 17^(6*n), 19^(6*n), 23^(3*n)..., etc. for K = 11, 2^(10*n), 3^(5*n), 5^(5*n), 7^(10*n), 13^(10*n), 17^(10*n), 19^(10*n), 29^(10*n)..., etc. for K = 13, 2^(12*n), 3^(3*n), 5^(4*n), 7^(12*n), 11^(12*n), 17^(6*n), 19^(12*n), 23^(6*n), 29^(4*n)..., etc. and so on... Last fiddled with by carpetpool on 20170407 at 02:19 
20170407, 02:15  #4  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Last fiddled with by science_man_88 on 20170407 at 02:16 

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