費(fèi)馬大定理推廣的一個(gè)特殊形式

2023-03-13 00:09 作者:我恨PDN定理 0人讀過 | 我要投稿

我們熟知的費(fèi)馬大定理，或稱為費(fèi)馬最后定理，是指以下的丟番圖方程無解： $x%5En%20%2By%5En%3Dz%5En%20%5C%20(n%3E2%20%5C%3A%E4%B8%94%5C%20xyz%5Cneq0%20)$

我們對(duì)這個(gè)式子進(jìn)行一步推廣，得到：

$a(x)%2Cb(x)%2Cc(x)%5Cin%20%5Cmathbb%7BC%7D%5C%2C%E4%B8%94%5C%2C%E4%B8%A4%5C%2C%E4%B8%A4%5C%2C%E4%BA%92%5C%2C%E7%B4%A0%0A%5C%5Ca%5En%2Bb%5En%5Cneq%20c%5En%20%5C%20(n%3E2)$

容易看出，當(dāng)三個(gè)多項(xiàng)式均為常數(shù)時(shí)，記為費(fèi)馬大定理的最初形式，我們這里假設(shè)三個(gè)多項(xiàng)式不為常數(shù)多項(xiàng)式. $%5Cbegin%7Balign*%7D%5Clabel%7B2%7D%0A%0A%26%20%5Cquad%20%E6%88%91%20%5C%2C%20%E4%BB%AC%20%5C%2C%20%E5%85%88%20%5C%2C%20%E7%BB%99%20%5C%2C%20%E5%87%BA%20%5C%2C%20%E9%9C%80%20%5C%2C%20%E8%A6%81%20%5C%2C%20%E7%9A%84%20%5C%2C%20%E5%BC%95%20%5C%2C%20%E7%90%86%20%5C%2C%20%E5%92%8C%20%5C%2C%20%E5%AE%9A%20%5C%2C%20%E7%90%86%20%5C%2C%20%2C%20%E8%AE%B0%20%5C%3B%20%20%20n_%7B0%7D(f(x))%20%20%20%5C%2C%20%E8%A1%A8%20%5C%2C%20%E7%A4%BA%20%5C%2C%20%20%20f(x)%20%5C%2C%20%20%20%E7%9A%84%20%5C%2C%20%E4%B8%8D%20%5C%2C%20%E5%90%8C%20%5C%2C%20%E5%A4%8D%20%5C%2C%20%E6%A0%B9%20%5C%2C%20%E4%B8%AA%20%5C%2C%20%E6%95%B0.%20%5C%5C%0A%26%20%E5%BC%95%20%5C%2C%20%E7%90%86%20%5Cquad%20%20%E8%8B%A5%20%5C%2C%20%20%20f(x)%2C%20g(x)%20%5Cin%20%5Cmathbb%7BC%7D%5Bx%5D%20%5Cbackslash%20%5Cmathbb%7BC%7D%20%2C%20%20%5C%2C%20%E5%88%99%5C%5C%0A%26%20%5Cquad%20(a)%3A%20%20%5Coperatorname%7Bdeg%7D%20f(x)%3D%5Coperatorname%7Bdeg%7D%5Cleft(f(x)%2C%20f%5E%7B%5Cprime%7D(x)%5Cright)%2Bn_%7B0%7D(f(x))%20%5C%5C%0A%26%20%5Cquad%20(b)%3A%20%20n_%7B0%7D(f(x)%20g(x))%20%5Cleq%20n_%7B0%7D(f(x))%2Bn_%7B0%7D(f(x))%20%2C%20%20%5C%2C%20%E7%AD%89%20%5C%2C%20%E5%8F%B7%20%5C%2C%20%E6%88%90%20%5C%2C%20%E7%AB%8B%20%5C%2C%20%E5%BD%93%20%5C%2C%20%20%20(f(x)%2C%20g(x))%3D1%20.%5C%5C%0A%26%E5%AE%9A%20%5C%2C%20%E7%90%86%20%5C%2C%20%20(Mason-Stothers)%20%5C%2C%20%20%E8%AE%BE%20%5C%2C%20%20%20f_%7B1%7D(x)%2C%20f_%7B2%7D(x)%2C%20f_%7B3%7D(x)%20%5Cin%20%5Cmathbb%7BC%7D%5Bx%5D%20%5Cbackslash%20%5Cmathbb%7BC%7D%20%2C%20%20%5C%2C%20%E8%8B%A5%20%5C%2C%20%20%20f_%7B1%7D(x)%2C%20f_%7B2%7D(x)%2C%20f_%7B3%7D(x)%20%20%5C%2C%20%20%E4%B8%A4%20%5C%2C%20%E4%B8%A4%20%5C%2C%20%E4%BA%92%20%5C%2C%20%E7%B4%A0%20%5C%2C%20%2C%20%20%5C%2C%20%E4%B8%94%5C%5C%0A%0A%26%20%5Cqquad%20f_%7B1%7D(x)%2Bf_%7B2%7D(x)%3Df_%7B3%7D(x)%5C%5C%0A%26%20%E5%88%99%5C%5C%0A%0A%26%20%5Cqquad%20%5Cmax%20%5Cleft%5C%7Bf_%7B1%7D(x)%2C%20f_%7B2%7D(x)%2C%20f_%7B3%7D(x)%5Cright%5C%7D%20%5Cleq%20n_%7B0%7D%5Cleft(f_%7B1%7D(x)%20f_%7B2%7D(x)%20f_%7B3%7D(x)%5Cright)-1%0A%5Cend%7Balign*%7D$

$%5Cbegin%7Balign%7D%5Clabel%7B2%7D%0A%26%20%E8%AF%81%5C%2C%20%E6%98%8E%5C%2C%20.%E6%98%BE%5C%2C%20%E7%84%B6%5C%2C%20%E6%88%91%5C%2C%20%E4%BB%AC%5C%2C%20%E6%9C%89%5C%5C%0A%26%20%5Cqquad%20%5Cbig%7B(%7D%20f_%7B3%7D(x)%2Cf'_%7B3%7D(x)%5Cbig%7B)%7D%5CBig%7B%7C%7D%5Cbig%7B(%7D%20f_%7B2%7D(x)f'_%7B3%7D(x)-f'_%7B2%7D(x)f_%7B3%7D(x)%5Cbig%7B)%7D%5C%5C%0A%26%20%5Cqquad%20%5Cbig%7B(%7D%20f_%7B2%7D(x)%2Cf'_%7B2%7D(x)%5Cbig%7B)%7D%5CBig%7B%7C%7D%5Cbig%7B(%7D%20f_%7B2%7D(x)f'_%7B3%7D(x)-f'_%7B2%7D(x)f_%7B3%7D(x)%5Cbig%7B)%7D%5C%5C%0A%26%20%5Cqquad%20%5Cbig%7B(%7D%20f_%7B1%7D(x)%2Cf'_%7B1%7D(x)%5Cbig%7B)%7D%5CBig%7B%7C%7D%5Cbig%7B(%7D%20f_%7B2%7D(x)f'_%7B1%7D(x)-f'_%7B2%7D(x)f_%7B1%7D(x)%5Cbig%7B)%7D%5C%5C%0A%26%20%5Cbecause%20f_%7B1%7D(x)%2Bf_%7B2%7D(x)%3Df_%7B3%7D(x)%20%5Cqquad%20%5Ctherefore%20f_%7B2%7D(x)f'_%7B3%7D(x)-f'_%7B2%7D(x)f_%7B3%7D(x)%3Df_%7B2%7D(x)f'_%7B1%7D(x)-f'_%7B2%7D(x)f_%7B1%7D(x)%5C%5C%0A%26%20%E4%BB%8E%5C%2C%20%E8%80%8C%5C%2C%20%20%7B%5Ctextstyle%20%5Cprod_%7Bi%3D1%7D%5E%7B3%7D%7D%5Cbig%7B(%7D%20f_%7Bi%7D(x)%2Cf'_%7Bi%7D(x)%5Cbig%7B)%7D%5CBig%7B%7C%7D%5Cbig%7B(%7D%20f_%7B2%7D(x)f'_%7B3%7D(x)-f'_%7B2%7D(x)f_%7B3%7D(x)%5Cbig%7B)%7D%5C%5C%0A%26%20%5Cbecause%20f_%7B2%7D(x)%5C%2C%20%E4%B8%8E%5C%2C%20%20f_%7B3%7D(x)%20%5C%2C%20%E4%BA%92%5C%2C%20%E7%B4%A0%5C%2C%20%2C%5C%2C%20%E7%94%B1%5C%2C%20(a)%5C%2C%20%E7%9F%A5%5C%5C%0A%26%20%5Cqquad%5Csum_%7Bi%3D1%7D%5E%7B3%7D%20%5Cbig%7B(%7D%20deg%5C%2Cf_%7Bi%7D(x)-n_%7B0%7D%5Cbig%7B(%7Df_%7Bi%7D(x)%5Cbig%7B)%7D%5Cbig%7B)%7D%5Cle%20deg%5C%2Cf_%7B2%7D(x)%2Bdeg%5C%2Cf_%7B3%7D(x)-1%5C%5C%0A%26%20%5Cbecause%20f_%7B1%7D(x)%2C%20f_%7B2%7D(x)%20%2C%20f_%7B3%7D(x)%20%5C%2C%20%E4%B8%A4%5C%2C%20%E4%B8%A4%5C%2C%20%E4%BA%92%5C%2C%20%E7%B4%A0%5C%2C%20%2C%5C%2C%20%E6%89%80%5C%2C%20%E4%BB%A5%5C%2C%20deg%5C%2Cf_%7B1%7D(x)%5Cle%20n_%7B0%7D(f_%7B1%7Df_%7B2%7Df_%7B3%7D)-1%5C%5C%0A%26%20%E5%90%8C%5C%2C%20%E7%90%86%5C%2C%20%E5%88%99%5C%2C%20deg%5C%2Cf_%7B2%7D(x)%5Cle%20n_%7B0%7D(f_%7B1%7Df_%7B2%7Df_%7B3%7D)-1%5C%5C%0A%26%20%E7%94%B1%5C%2C%20f_%7B1%7D(x)%2Bf_%7B2%7D(x)%3Df_%7B3%7D(x)%5C%2C%20%E7%9F%A5%5C%2C%20deg%5C%2Cf_%7B3%7D(x)%5Cle%20n_%7B0%7D(f_%7B1%7Df_%7B2%7Df_%7B3%7D)-1%5C%5C%0A%26%20%E4%BB%8E%5C%2C%20%E8%80%8C%5C%2C%20%E5%AE%9A%5C%2C%20%E7%90%86%5C%2C%20%E5%BE%97%5C%2C%20%E8%AF%81%5C%2C%20.%5C%5C%0A%26%20%E5%9B%9E%5C%2C%E5%88%B0%5C%2C%E5%8E%9F%5C%2C%E9%A2%98%5C%2C.%5C%5C%0A%26%20%5Cqquad%20%E8%8B%A5%5C%2C%5Cexists%20%5C%2Cf_%7B1%7D(x)%2Cf_%7B2%7D(x)%2Cf_%7B3%7D(x)%5C%2C%E7%AC%A6%5C%2C%E5%90%88%5C%2C%E9%A2%98%5C%2C%E6%84%8F%5C%2C%2C%5C%2C%E5%88%99%5C%5C%0A%26%20%5Cqquad%20ndeg%5C%2Cf_%7Bi%7D(x)%3Ddeg%5C%2Cf_%7Bi%7D%5En(x)%20%5Cle%20n_%7B0%7D(f_%7B1%7D%5Enf_%7B2%7D%5Enf_%7B3%7D%5En)-1%3Dn_%7B0%7D(f_%7B1%7Df_%7B2%7Df_%7B3%7D)-1%5C%5C%0A%26%20%5Cqquad%20%5Ctherefore%20n%5Csum_%7Bi%3D1%7D%5E%7B3%7D%20deg%5C%2Cf_%7Bi%7D(x)%5Cle%203n_%7B0%7D(f_%7B1%7Df_%7B2%7Df_%7B3%7D)-3%5C%5C%0A%26%20%5Cqquad%20%5Ctherefore%20(n-3)%5C%2Cdeg%5C%2C(f_%7B1%7Df_%7B2%7Df_%7B3%7D)%20%5Cle%20-3%5C%5C%0A%26%20%E7%94%B1%5C%2Cdeg%5C%2C(f_%7B1%7Df_%7B2%7Df_%7B3%7D)%5Cge%203%2C%5C%2C%E5%8F%AF%5C%2C%E5%BE%97%5C%2Cn%5Cle2%5C%5C%0A%26%20%E7%9F%9B%5C%2C%E7%9B%BE%5C%2C%EF%BC%81%5C%2C%E6%95%85%5C%2C%E5%8E%9F%5C%2C%E5%91%BD%5C%2C%E9%A2%98%5C%2C%E6%88%90%5C%2C%E7%AB%8B%5C%2C.%5CBox%20%0A%5Cend%7Balign%7D$

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費(fèi)馬大定理推廣的一個(gè)特殊形式

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