一类超椭圆方程的整数解

The Integer Solutions of a Class of Superelliptic Equations

设ｍ，ｎ∈Ｎ；ｍ≥２，ｎ≥２，ｍｎ≥６，ｆ（ｘ）＝ｘｍ＋ａ１ｘｍ－１＋…＋ａｍ∈Ｚ［ｘ］，Ｈ＝ｍａｘ（｜ａ１｜，…，｜ａｍ｜）．本文运用组合分析方法证明了：当ｍ≡０（ｍｏｄｎ），ａ１，…，ａｍ不全为零，而且其中第一个非零系数ａｓ与ｎ互素时，方程ｆ（ｘ）＝ｙｎ，ｘ，ｙ∈Ｚ，仅有有限多组解（ｘ，ｙ）。

Let m, n ∈ N such that m≥2 , n≥2 and mn≥6. Let f(x) = xm+a,xm-1+''' +am ∈ Z[x] with as ≠ 0 and aj=0 (1 ≤ j < s), and let In this paper,with some combinatorial analysis methods, we prove that if m≡0(mod n) and gcd (as, n)=1,then the equation f(x) = yn has only finitely many integer solutions (x,y). Moreover, allsolutions (x, y) of the equation satisfy |x| < (4mH)2m/n+1 and |y|< (4mH)2m2/n2+m/n+1.