| 摘要: |
| 设n>4,fb(x)=xn-bx-a∈Z[x],其中a,b≠0,n∈N,a,b∈Z.讨论b=±1时fb(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. |
| 关键词: 多项式 二次不可约因式 本原素因数 整系数 Lucas数 |
| DOI： |
| 投稿时间：2004-08-04修订日期：2004-10-12 |
| 基金项目:四川省教育厅自然科学基金(2004B025)资助项目。 |
|
| The Irreducible Quadratic Factorizations of the Polynomial xn-bx-a |
|
He Bo
|
| (Longchang Xiangshi Middle School, Longchang, Sichuan, 642152, China) |
| Abstract: |
| Let n>4,fb(x)=xn-bx-a∈Z with a,b≠0,n∈N,a,b∈Z.We have discussed the irreducible quadratic factorizations of the polynomial fb(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. |
| Key words: polynomial irreducible quadratic factorizations primitive divisors intergral coefficient Lucas numbers |
