School:李煌數學研究院/举出一个非平凡存在根式解的代数方程

李煌定理

已知费马方程$a^n+b^n=c^n$成立

则等式$c^{7n}=\frac{1}{16}(-2(-c^{4n}(8c^n+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12)b^n+$ $3c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4)b^{2n}+$$2c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^{3n}-$$20c^{4n}(-6c^n+c^{2n}-9)b^{4n}+24c^{2n}(-4c^n+3c^{2n}-6)b^{5n}-$$112c^{2n}{b}^{6n}+64b^{7n}+c^{6n}(2c^{2n}+6c^{2n}+7c^{3n}-7c^{4n}-6c^{5n}+3))a^n+$$(-6c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4)b^n-$$6c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^{2n}+80c^{4n}$$(-6c^n+c^{2n}-9)b^{3n}-120c^{2n}(-4c^n+3c^{2n}-6)b^{4n}+672c^{2n}{b}^{5n}-$$448b^{6n}+c^{4n}(8c^n+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12))a^{2n}-$$2(2c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^n-$$40c^{4n}(-6c^n+c^{2n}-9)b^{2n}+80c^{2n}(-4c^n+3c^{2n}-6)b^{3n}-$$560c^{2n}{b}^{4n}+448b^{5n}+c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4))a^{3n}-$$(-40c^{4n}(-6c^n+c^{2n}-9)b^n+120c^{2n}(-4c^n+3c^{2n}-6)b^{2n}-1120c^{2n}{b}^{3n}+$$1120b^{4n}+c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8))a^{4n}-$$8(6c^{2n}(-4c^n+3c^{2n}-6)b^n-84c^{2n}{b}^{2n}+112b^{3n}+c^{4n}(6c^n-c^{2n}+9))a^{5n}-$$8(-28c^{2n}{b}^n+56b^{2n}+c^{2n}(-4c^n+3c^{2n}-6))a^{6n}-$$32(4b^n-c^{2n})a^{7n}-16a^{8n}+32b^{7n}{c}^{2n}-$$16b^{8n}+8b^{5n}{c}^{4n}(-6c^n+c^{2n}-9)-$$8b^{6n}{c}^{2n}(-4c^n+3c^{2n}-6)+$$2b^n{c}^{6n}(-2c^n-6c^{2n}-7c^{3n}+7c^{4n}+6c^{5n}-3)+$$c^{6n}(6c^n+2c^{2n}-4c^{3n}+27c^{4n}+6c^{5n}-9c^{6n}-4)-$$b^{4n}{c}^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)+$$b^{2n}{c}^{4n}(8c^n+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12)+$$b^{3n}(8c^{4n}+52c^{6n}+56c^{7n}-14c^{8n}-4c^{9n}))$

也成立

参考

• 《南昌理工學院學報》.李煌

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