Formulas for Coefficients of Hundred Cyclotomics and Numbers Battle

Abstract

In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index n in x i . We will only determine 1003 for an index n odd and a degree . The others will be deduced, we’ll see how. The formula is , without exception if u(n)=-1 or if 4 doesn’t divide and with its 165 exceptions of which 7 when u(n)=0 and 158 when u(n)=1 that will be shared in 154 and 4 pairs (n, i), which we will specify the conditions and values of the coefficients. According to u(n), according to the class of i modulo p, the first factor of the prime factor decomposition of n when u(n)=1 and according to gcd(n, i), the formula will or will not be valid and replaced otherwise by the good value that will be 0 for 152 pairs (n,i) or 1 in the 13 other exceptions.