Formulas for Coefficients of Hundred Cyclotomics and Numbers Battle

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 i x . We will only determine 1003 for an index n odd and a degree ( ) 2 n i ϕ ≤ . The others will be deduced, we’ll see how. The formula is ( ) ( ) ; , n coe n i gcd n i µ  = −       , without exception if ( ) 1 n µ = − or if 4 doesn’t divide ( ) n ϕ and with its 165 exceptions of which 7 when ( ) 0 n µ = and 158 when ( ) 1 n µ = that will be shared in 154 and 4 pairs ( ) , n i , which we will specify the conditions and values of the coefficients. According to ( ) n µ , according to the class of i modulo p, the first factor of the prime factor decomposition of n when ( ) 1 n µ = 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.


Introduction
Note that formula numbers will be in parentheses (x), while bibliographic references will be enclosed in square brackets [y].
.Ah! This cyclotomic world and this "cyclotomic" word that one does not even find in a normal dictionary (You need an encyclopedia) and which recalls cycling and filming, but which also inspires this apparatus which emits light rays of which the rays are reduced and enlarged and stun the voyeur, the observer and the contemplator. Thus is this theory of cyclotomic polynomials with its descents and ascents, in their degrees, their coefficients, their numbers, their real and complex values, their roots, etc.
Yes finally formulas for cyclotomics and their coefficients, as indicated by the title. Formulas that we will try to demonstrate and verify by examples and scripts with maple 12 in principle for the first hundred, except indication and precision for a higher index.
The main objective of this article is to establish the formula of the theorem 3.9, the theorem 3.13 and its corollary 3.15.
Like parraph 6 which will make it possible to determine all the cyclotomics thanks to those of prime indices whose expression is known (formula (6)). The goal is also to find practical and easy formulas applicable to the hand and without data processing, for the cyclotomics and their coefficients, until nowadays nonexistent. All we know today is that the first distinct coefficient of −1, 0 and 1 is at the degree i = 7 and degree j = 41 of cyclotomic index n = 105. [2]. In short, what we want is: A formula for calculating the coefficients. An easy method to designate cyclotomic polynomials. Develop the cyclotomic theory, spread it and make it more touchable, more ready and more popular among mathematicians.
The article, its presentation and the demonstrations of the results are not always classical, they are sometimes done by checking with Maple scripts, especially that these results concern in general only the first hundred cyclotomics, but which will be able to inspire generalization in the future. It is the case in this article for the proposition 3.3 and the theorem 3.13 that we have extended up to 1000 only (but the formula is still true) because the script would require a big delay and could block execution if it is extended more. we know that today there are no formulas for these coefficients, not even for a finite number of them. This article is unpublished, new and never published.
We will adopt in the text and with maple 12, the notation ϕ for the indicator of Euler phi and ( ) cy n to designate the cyclotomic polynomial of index n of indefinite x by default, otherwise we will specify another indeterminate y we want to assign to x by ( ) , cyc n y , (y can be x − , 2 x , 3 x or any other power of x). We will denote by ( ) , coe n i the coefficient of ( ) cy n in i x , All this after having downloaded the "numtheory" package.
Thus we will have for example: In polynomial theory, the n th cyclotomic polynomial is the unit polynomial whose roots are the primitive roots n th of the unit. It is of degree ( ) n ϕ [3] (See Serge Lang [3]), indicator function of Euler ϕ that we will define, It is with integer coefficients and is irreducible on  . For any integer m, the polynomial 1 m X − is the product of the cyclotomic polynomials associated with the divisors of m (see [4] , which at k associates 2 π e ik n . So a root n th , 2 e ik n ⋅Π will generate n G (that is, its powers will cover n G ) if and only if k generates the additive group n ⋅   (that is, its multiples will encompass n ⋅  ), which equals k first with n [5] (see [5]).
The number of non-zero k integers strictly less than k is ( ) n ϕ , which is why the degree of ( ) , n x ϕ , the n th cyclotomic polynomial in x is ( ) n ϕ , this Euler's indicator, whose properties will be studied, which will involve others on cyclotomic polynomials. With maple, and to simplify the notations, we will define the function cy which at an integer n will associate the n th cyclotomic noted in maple by We thus have The analysis of these polynomials allows the resolution of many problems. As an example of cyclotomic polynomials: 1 cy x x x x = + + + + and ( ) 2 6 1 cy x x = − + Definition 2.3. The function ϕ , indicator of Euler is the one which to an integer n associates the number of primes with n strictly lower than n. Remark 2.4. This is also the number of generators of n G , the multiplicative group of n th roots of 1. Definition 2.5. The cyclotomic polynomial of index n is the polynomial whose roots are the generators of G n . Property 2.6.
That is, we have a symmetry of the coefficients of ( ) cy n with respect to that of ( ) 2 n x ϕ . Definition 2.7. We call function of Mobius µ , the function which at a positive integer n associates ( ) n µ with ( ) 0 n µ = if n is with square factor, that is, if it is divisible by the square of a prime number, otherwise it's 1 if n is divisible by the even of a prime numbers; or −1 if the number of prime numbers that divide it is odd.
Recall that ( ) , gcd n i denotes with maple the gcd of n and i. We will define in the same way that cy and cyc the function , values that will be used to set conditions of validity or not of certain formulas. And for the last paragraphs we will also need ( ) 2 , mob n i which will express the largest of ( ) is known (It is 1 x − ) and we know that the constant coefficient of a cyclotom polynomial is 1. This will avoid us to specify ("from 0"), since without this precision, maple starts by default the variation of a parameter starting from 1, when one does not specify the first value. By against it is necessary to him to specify the last one with the help of to " ( ) 2 n ϕ ".
We will deduce the ones for the n pairs by the formulas we will recall linking the ( )

Formula for the Coefficients
First, remember the following properties: 2) on  two cyclotomic polynomials are prime between them. In this article, we will try to solve the riddle of the cyclotomic coefficients, knowing that we have the following remarks and comments: a) If n is prime then the nth cyclotomic polynomial is To see the difference between the formulas of (2) and (5) when p is prime and divide or not divide n we have as an example for ( ) cy pn : We have a symmetry around 24 x , since ( ) −2 is for example coefficient of 7 x and of 48 7 We'll just make this proposal using a maple script for n up to 1000.     Note that we have , mob n i in 152 and we never have the opposite. Verification scripts will not be prepared for this, but they are similar to those that will be established.
is not multiple of 4, then except for There are 16 to 100 and 98 until 1000.
Let's check with the following maple script and its counter k which will count the number of ( )

Other Properties of Cyclotomic Coefficients
1) Up to 104 n = , the coefficients of ( ) cy n are always −1, 0 or 1, and it is only from 105 that the first coefficient appears-2 [7] (see [7]). with maple we will also check.
2) Up to 384, a coefficient of ( ) cy n is never an absolute value strictly greater than 2, with maple, we can see that, as well as the following properties.

( )
cy n has all its coefficients in  and it is irreducible on  , so on  (see [5]). And on a nonzero characteristic field, if ( ) cy n is irreducible, then ( ) cy d is for all d divisor of n (see [4]). We also have that for all m ∈  , there exists n such that m or -m is coefficient of ( ) cy n (see [8]), for all 1 n > there is a symmetry of the coefficients of ( ) cy n around ( ) 2 n ϕ see [2], and if n is prime, then ( ) cy n is

Prime Indices and Determination of a Cyclotomic with All Its Coefficients
Again and again cyclotomics. This time for the direct calculation of cyclotomics.
For that, let's remember that we will adopt the notations in maple 12 that follow, after loading the package "with (numtheory)" (mul is the product): The first need of a cyclotomic of index a prime number p, to calculate another is to calculate its square.
The second need is to calculate the one of index its product by the first number which follows it, if one does not want to use that of index that which follows it, and if one wants always to use the smallest possible ones.
For example in the following scripts, we will use ( ) If we wanted to go further, we would have needed the index 11, that when we arrived at 11 2 , i.e. 121, then for 11 13 143 × = . So we will only use 2 th and 3 'em cyclotomic up to ( ) 24 cy .
-we will only use the 2 th , 3 th and 5 th cyclotomics up to ( ) 48 cy .
we will only need 2 nd , 3 th , 5 th and 7 th for all our first 100 cyclotomics, and we could have advanced up to to 120, since the next prime number 7 is 11, and since ( ) So let's go for some examples: 1 Whereas for cy(105) (resp. cy(385)), the first product of three distinct prime numbers two to two, we find the first coefficients of absolute value strictly greater than 1, in degrees 7 and 41, symmetrically compared to ( ) 48 105 ϕ = (resp.strictly greater than 2, in 119, 120 and 121, which is ( )

Examples
We will give two examples of cyclotomics with indices greater than 100 to determine and to determine their coefficients, to see how we reduce ourselves to a cyclotomic index less than 100. It will be cy(225) and cy(120).  , which is 1 and −1.
However, for 2 a which is part of the 53, the coefficient is 0 because

Conclusion
We hope to have raised some riddles of cyclotomic polynomials and their coefficients, and the enigma that has pained many mathematicians and researchers, this by the formula F12 and by means of prime numbers as in the examples of paragraph 6. Maple, Euler and Mobius helped us a lot. It's not rocket science. We will advance in future articles.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.