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ysr

3^3+4^3+5^3=6^3
各项居然是连续的整数，类似的等式还又吗，成等差数列的也可以？

Treenewbee

ysr 发表于 2024-3-14 08:44
3^3+4^3+5^3=6^3
各项居然是连续的整数，类似的等式还又吗，成等差数列的也可以？

\[(n - k)^3 + n^3 + (n + k)^3 = (n + 2 k)^3\rightarrow(n-4 k) \left(n^2+k n+k^2\right)=0\rightarrow n=4k\]

蔡家雄

求：\(x^{2n}+y^{2n}+z^{2n+2}=w^{2n+1}\)

求：\(x^{2n}+y^{2n+2}+z^{2n+2}=w^{2n+1}\)

蔡家雄

求：\(x^{6}+y^{6}+z^{10}=w^{14}\)

蔡家雄

求：\(x^{6}+y^{10}+z^{10}=w^{14}\)

Treenewbee

蔡家雄 发表于 2024-3-17 17:53
求：\(x^{6}+y^{6}+z^{10}=w^{14}\)

\[\left(2^{21} 3^{25}\right)^6+\left(2^{22} 3^{25}\right)^6+\left(2^{13} 3^{15}\right)^{10}=\left(2^9 3^{11}\right)^{14}\]

Treenewbee

蔡家雄 发表于 2024-3-17 17:53
求：\(x^{6}+y^{10}+z^{10}=w^{14}\)

\[\left(2^{28} 3^{30}\right)^6+\left(2^{17} 3^{18}\right)^{10}+\left(2^{17} 3^{18}\right)^{10}=\left(2^{12} 3^{13}\right)^{14}\]

Treenewbee

蔡家雄 发表于 2024-3-17 20:25
求：\(x^a - y^b=2\) 的正整数解，

{3,3,5,2}
{n^r+2,1,n,r}

Treenewbee

Treenewbee 发表于 2024-3-18 08:05
\[\left(2^{21} 3^{25}\right)^6+\left(2^{22} 3^{25}\right)^6+\left(2^{13} 3^{15}\right)^{10}=\left( ...

小一点的解：\[\left(2^{12} 3^{30}\right)^6+\left(2^{12} 3^{30}\right)^6+\left(2^7 3^{18}\right)^{10}=\left(2^5 3^{13}\right)^{14}\]

蔡家雄

求：\(x^{6}+y^{10}=2*z^{14}\)