求方程x^3-y^3+x^2y-xy^2=32的正整数解RT

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求方程x^3-y^3+x^2y-xy^2=32的正整数解RT求方程x^3-y^3+x^2y-xy^2=32的正整数解RT求方程x^3-y^3+x^2y-xy^2=32的正整数解RTx³-y&

求方程x^3-y^3+x^2y-xy^2=32的正整数解RT
求方程x^3-y^3+x^2y-xy^2=32的正整数解
RT

求方程x^3-y^3+x^2y-xy^2=32的正整数解RT
x³-y³+x²y-xy²=x²(x+y)-y²(x+y)
=(x²-y²)(x+y)
=(x-y)(x+y)²=32
x,y都是正整数∴x+y≥2,（x+y）²≥4,
而32=2*2*2*2*2
32的因数中为完全平方数且大于等于4的只有4,16
若（x+y）²=4
则x=y=1,x-y= 0,明显不成立
∴（x+y）²=16
即x+y=4
∴x-y=2
即 x=3,y=1 为原方程唯一一对正整数解

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