多项式xn-bx-a的二次不可约因式

The Irreducible Quadratic Factorizations of the Polynomial xn-bx-a

关键词:  多项式  二次不可约因式  本原素因数  整系数  Lucas数

Key words:  polynomial  irreducible quadratic factorizations  primitive divisors  intergral coefficient  Lucas numbers

摘要:

设n>4,f_b(x)=xⁿ-bx-a∈Z[x],其中a,b≠0,n∈N,a,b∈Z.讨论b=±1时f_b(x)的二次不可约因式.证明:x⁶-x-a在Z[x]中没有二次不可约因式;若f_-1(x)在Z[x]中有二次不可约因式,除了n≡2(mod 3),a=-1,g(x)=x²+x+1情况外,必有n=5,a=±6或n=13,a=±90,且g(x)=x²±x+2.

Abstract:

Let n>4,f_b(x)=xⁿ-bx-a∈Z with a,b≠0,n∈N,a,b∈Z.We have discussed the irreducible quadratic factorizations of the polynomial f_b(x) with b=±1.We proved that:x⁶-x-a has not irreducible quadratic factorizations in Z[x];f_-1(x) has an irreducible quadratic factorization g(x) in Z[x] witch is monic,then either n≡2(mod 3),a=-1,g(x)=x²+x+1,or n=5,a=±6,or n=13,a=±90,g(x)=x²±x+2.