摘要
对每个奇素数p,我们提出了关于p的两个命题:(I)不存在正整数m和n,使代数曲线f(y)=y^3-(2m+1)y-2n的三个零点同时为p方有理整数;(I)不存在正整数m,使不定方程x^2+3y^2=4(2m+1)具有三组正整数解x_i,y_i(i=1,2,3),满足x_1=x_2+x_3,x_1~2+x_2~2+x_3~2=6(2m+1)。本文旨在证明(I)及(I)都与关于素数p的Fermat猜想等价。
For every odd prime p,two propositions on p are given in this paper.One of them is( I ).there are no positive integers m and n so that the three ziros of the algebric curve f(y) = y - (2m + l)y - 2n are all the form ap i.e. p powers of integers; the other is ( II ) . there is no positive integer m so that Diophantion equation x2 + 3y2p = 4(2m + 1) has three positive integer solutions xi, yi(i= 1,2,3 ) which can satisfy the following conditions--x1=x2 +x3,
X12 + x22 + X32 = 6(2m + 1). The two propositions are all equivalents to the Ferm-at's Conjecture on odd prime p.
关键词
Fermat猜想
多项式
零点
Fermat's Conjecture
Zero of a polymomial
A new class of Diophantion equation
