# Why does the coefficient value of the binary polynomial x ^ n-1 appear other than –1,1?

I factorize the binomial polynomial x ^ n-1 and observe the coefficients of the arguments. I thought that the coefficients of the arguments of x ^ n-1 would all be -1,1, but in fact they did not. For example, a factor of 2 appears in the coefficient of the argument of x ^ 105-1.

As the value of n increases, the coefficients of the arguments start to appear as well as 2, 3, 4, 5, ..., and so on. The following figure shows the table of the formulas and the number of times the coefficients appear first, from n = 1 to n = 10000.

I wonder if this phenomenon is known to the cause and the rules. Is there a mathematical reason for the n value where other coefficients than 1, -1 appear? Or is it an irregular phenomenon? Thank you for letting me know about the research article, site or contents.

• what the heck! i did not expect that.. – Saketh Malyala Jun 9 '17 at 5:06
• These are the cyclotomic polynomials. The fact that $105$ is the first with a factor that is not $\pm 1$ is because it is the product of three odd primes. If you search coefficients cyclotomic polynomials you will find a lot. – Ross Millikan Jun 9 '17 at 5:11
• Thank you for answer. What I did not know. I will study the part of what you give me. – user21427 Jun 9 '17 at 5:43

The polynomial $$x^n-1=\prod_{d\mid n}\Phi_d(x)$$ where $\Phi_d$ is the $d$-th cyclotomic polynomial. This is the polynomial whose zeroes are precisely the primitive $d$-th roots of unity. Yes, $\Phi_{105}$ has a coefficient $\pm2$. It is known that arbitrarily large coefficients do occur in the cyclotomic polynomials.