一题 \(x^{131}+y^{137}+z^{139}=w^{149}\) 多解

ysr

蔡家雄 发表于 2025-3-30 12:57
虽然 \(x^2 - 13^2*17*y^2= -1\) 有 正整数解，，

但是 \(x^2 - 13*17^2*y^2= -1\) 无 正整数解，，

x^2-13^2*17y^2=-1的解：
x=268  y=5
x=76996132  y=1436485

5.261719秒是100000000内的解

ysr

蔡家雄 发表于 2025-3-29 23:48
求 \(x^2 - (17*25)*y^2= -1\) 的正整数解，，

x^2-17*25y^2=-1的解：
x=268  y=13
x=76996132  y=3734861

5.257813秒是10000 0000 内的解

ysr

蔡家雄 发表于 2025-3-29 23:49
求 \(x^2 - (29*37)*y^2= -1\) 的正整数解，，

x^2-37*29y^2=-1的解：
x=45368  y=1385

5.318359秒是100000000内的解

ysr

蔡家雄 发表于 2025-3-29 23:51
求 \(x^2 - (53*61)*y^2= -1\) 的正整数解，，

x^2-3233y^2=-1的解：
x=1065376  y=18737

5.439453

ysr

蔡家雄 发表于 2025-4-2 01:44
同时 有解的 佩尔方程，

设 \(4k+1=a^2+16^2=b^2+44^2\) ，

答案以此为：
x=44  y=1
x=340868  y=7745

5.445313

2105这个在10000 0000内无解，可能解不存在

x=34208  y=689

5.45703

x=173932  y=2525

5.417969

8177这个在10000 0000内无解，可能解不存在

12137这个在10000 0000内无解，可能解不存在

20705这个在10000 0000内无解，可能解不存在

177497这个在10000 0000内无解，可能解不存在

蔡家雄

同时 有解的 佩尔方程，

设 \(4k+1=a^2+8^2=b^2+32^2\) ，

则 \(4k+1=31^2+8^2 , 53^2+8^2 , 83^2+8^2 , 241^2+8^2\) 这四个数值，

则 \(x^2 -(4k+1)*y^2= -1\) 都有 正整数解，，

得 \(4k+1=31^2+8^2\) 的最小解：\(x=32\) , \(y=0^2+1^2\) ,

得 \(4k+1=53^2+8^2\) 的最小解：\(x=268\) , \(y=1^2+2^2\) ,

得 \(4k+1=83^2+8^2\) 的最小解：\(x=1084\) , \(y=2^2+3^2\) ,

得 \(4k+1=241^2+8^2\) 的最小解：\(x=27248\) , \(y=7^2+8^2\) ,

ysr

蔡家雄 发表于 2025-4-2 12:17
同时 有解的 佩尔方程，

设 \(4k+1=a^2+8^2=b^2+32^2\) ，

31*31+64=1025解为：
x=32  y=1
x=131168  y=4097

5.390625

53*53+64=2873解为：
x=268  y=5
x=76996132  y=1436485

5.375

83*83+64=6953解为：
x=1084  y=13

5.382813

241*241+64=58145解为：
x=27248  y=113

5.382813

蔡家雄

同时 无解的 佩尔方程，

设 \(4k+1=a^2+4^{2t}=b^2+4^{2t+2}\) ，

则 \(x^2 -(4k+1)*y^2= -1\) 都无 正整数解，，


同时 有解的 佩尔方程，

设 \(4k+1=a^2+4^{2t+1}=b^2+4^{2t+3}\) ，

则 \(x^2 -(4k+1)*y^2= -1\) 都有 正整数解，，


同时 有解的 佩尔方程，

设 \(4k+1=a^2+4^{t+1}=b^2+4^{t+2}\) ，

则 \(x^2 -(4k+1)*y^2= -1\) 都有 正整数解，，

Treenewbee

蔡家雄 发表于 2025-4-3 08:57
求 \(x^2 - 13*y^2=314159\) 的正整数解，

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Treenewbee

蔡家雄 发表于 2025-4-3 08:57
求 \(x^2 - 17*y^2=314159\) 的正整数解，

{{1228,265},
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