已知费马方程 a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} 成立
则等式 c 7 n = 1 16 ( − 2 ( − c 4 n ( 8 c n + 22 c 2 n + 28 c 3 n − 41 c 4 n − 38 c 5 n + 2 c 6 n + 12 ) b n + {\displaystyle c^{7n}={\frac {1}{16}}(-2(-c^{4n}(8c^{n}+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12)b^{n}+} 3 c 4 n ( − 26 c 2 n − 28 c 3 n + 7 c 4 n + 2 c 5 n − 4 ) b 2 n + {\displaystyle 3c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4)b^{2n}+} 2 c 2 n ( 52 c 2 n + 56 c 3 n − 44 c 4 n − 24 c 5 n + c 6 n + 8 ) b 3 n − {\displaystyle 2c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^{3n}-} 20 c 4 n ( − 6 c n + c 2 n − 9 ) b 4 n + 24 c 2 n ( − 4 c n + 3 c 2 n − 6 ) b 5 n − {\displaystyle 20c^{4n}(-6c^{n}+c^{2n}-9)b^{4n}+24c^{2n}(-4c^{n}+3c^{2n}-6)b^{5n}-} 112 c 2 n b 6 n + 64 b 7 n + c 6 n ( 2 c 2 n + 6 c 2 n + 7 c 3 n − 7 c 4 n − 6 c 5 n + 3 ) ) a n + {\displaystyle 112c^{2n}b^{6n}+64b^{7n}+c^{6n}(2c^{2n}+6c^{2n}+7c^{3n}-7c^{4n}-6c^{5n}+3))a^{n}+} ( − 6 c 4 n ( − 26 c 2 n − 28 c 3 n + 7 c 4 n + 2 c 5 n − 4 ) b n − {\displaystyle (-6c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4)b^{n}-} 6 c 2 n ( 52 c 2 n + 56 c 3 n − 44 c 4 n − 24 c 5 n + c 6 n + 8 ) b 2 n + 80 c 4 n {\displaystyle 6c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^{2n}+80c^{4n}} ( − 6 c n + c 2 n − 9 ) b 3 n − 120 c 2 n ( − 4 c n + 3 c 2 n − 6 ) b 4 n + 672 c 2 n b 5 n − {\displaystyle (-6c^{n}+c^{2n}-9)b^{3n}-120c^{2n}(-4c^{n}+3c^{2n}-6)b^{4n}+672c^{2n}b^{5n}-} 448 b 6 n + c 4 n ( 8 c n + 22 c 2 n + 28 c 3 n − 41 c 4 n − 38 c 5 n + 2 c 6 n + 12 ) ) a 2 n − {\displaystyle 448b^{6n}+c^{4n}(8c^{n}+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12))a^{2n}-} 2 ( 2 c 2 n ( 52 c 2 n + 56 c 3 n − 44 c 4 n − 24 c 5 n + c 6 n + 8 ) b n − {\displaystyle 2(2c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)b^{n}-} 40 c 4 n ( − 6 c n + c 2 n − 9 ) b 2 n + 80 c 2 n ( − 4 c n + 3 c 2 n − 6 ) b 3 n − {\displaystyle 40c^{4n}(-6c^{n}+c^{2n}-9)b^{2n}+80c^{2n}(-4c^{n}+3c^{2n}-6)b^{3n}-} 560 c 2 n b 4 n + 448 b 5 n + c 4 n ( − 26 c 2 n − 28 c 3 n + 7 c 4 n + 2 c 5 n − 4 ) ) a 3 n − {\displaystyle 560c^{2n}b^{4n}+448b^{5n}+c^{4n}(-26c^{2n}-28c^{3n}+7c^{4n}+2c^{5n}-4))a^{3n}-} ( − 40 c 4 n ( − 6 c n + c 2 n − 9 ) b n + 120 c 2 n ( − 4 c n + 3 c 2 n − 6 ) b 2 n − 1120 c 2 n b 3 n + {\displaystyle (-40c^{4n}(-6c^{n}+c^{2n}-9)b^{n}+120c^{2n}(-4c^{n}+3c^{2n}-6)b^{2n}-1120c^{2n}b^{3n}+} 1120 b 4 n + c 2 n ( 52 c 2 n + 56 c 3 n − 44 c 4 n − 24 c 5 n + c 6 n + 8 ) ) a 4 n − {\displaystyle 1120b^{4n}+c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8))a^{4n}-} 8 ( 6 c 2 n ( − 4 c n + 3 c 2 n − 6 ) b n − 84 c 2 n b 2 n + 112 b 3 n + c 4 n ( 6 c n − c 2 n + 9 ) ) a 5 n − {\displaystyle 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 ( − 28 c 2 n b n + 56 b 2 n + c 2 n ( − 4 c n + 3 c 2 n − 6 ) ) a 6 n − {\displaystyle 8(-28c^{2n}b^{n}+56b^{2n}+c^{2n}(-4c^{n}+3c^{2n}-6))a^{6n}-} 32 ( 4 b n − c 2 n ) a 7 n − 16 a 8 n + 32 b 7 n c 2 n − {\displaystyle 32(4b^{n}-c^{2n})a^{7n}-16a^{8n}+32b^{7n}c^{2n}-} 16 b 8 n + 8 b 5 n c 4 n ( − 6 c n + c 2 n − 9 ) − {\displaystyle 16b^{8n}+8b^{5n}c^{4n}(-6c^{n}+c^{2n}-9)-} 8 b 6 n c 2 n ( − 4 c n + 3 c 2 n − 6 ) + {\displaystyle 8b^{6n}c^{2n}(-4c^{n}+3c^{2n}-6)+} 2 b n c 6 n ( − 2 c n − 6 c 2 n − 7 c 3 n + 7 c 4 n + 6 c 5 n − 3 ) + {\displaystyle 2b^{n}c^{6n}(-2c^{n}-6c^{2n}-7c^{3n}+7c^{4n}+6c^{5n}-3)+} c 6 n ( 6 c n + 2 c 2 n − 4 c 3 n + 27 c 4 n + 6 c 5 n − 9 c 6 n − 4 ) − {\displaystyle c^{6n}(6c^{n}+2c^{2n}-4c^{3n}+27c^{4n}+6c^{5n}-9c^{6n}-4)-} b 4 n c 2 n ( 52 c 2 n + 56 c 3 n − 44 c 4 n − 24 c 5 n + c 6 n + 8 ) + {\displaystyle b^{4n}c^{2n}(52c^{2n}+56c^{3n}-44c^{4n}-24c^{5n}+c^{6n}+8)+} b 2 n c 4 n ( 8 c n + 22 c 2 n + 28 c 3 n − 41 c 4 n − 38 c 5 n + 2 c 6 n + 12 ) + {\displaystyle b^{2n}c^{4n}(8c^{n}+22c^{2n}+28c^{3n}-41c^{4n}-38c^{5n}+2c^{6n}+12)+} b 3 n ( 8 c 4 n + 52 c 6 n + 56 c 7 n − 14 c 8 n − 4 c 9 n ) ) {\displaystyle b^{3n}(8c^{4n}+52c^{6n}+56c^{7n}-14c^{8n}-4c^{9n}))}
也成立
<<School:李煌数学研究院
