用公式法求解特殊佩尔方程

cz1

三個三位數相加之和是2024，如果 9 個數字並無重複，可以有解吗 ？

Treenewbee

cz1 发表于 2024-2-9 09:59
三個三位數相加之和是2024，如果 9 個數字並無重複，可以有解吗 ？

360个解

{{203,845,976},{203,846,975},{203,854,967},{203,857,964},{203,864,957},{203,867,954},{203,875,946},{203,876,945},{204,853,967},{204,857,963},{204,863,957},{204,867,953},{205,843,976},{205,846,973},{205,873,946},{205,876,943},{206,843,975},{206,845,973},{206,873,945},{206,875,943},{207,853,964},{207,854,963},{207,863,954},{207,864,953},{243,805,976},{243,806,975},{243,875,906},{243,876,905},{245,803,976},{245,806,973},{245,873,906},{245,876,903},{246,803,975},{246,805,973},{246,873,905},{246,875,903},{253,804,967},{253,807,964},{253,864,907},{253,867,904},{254,803,967},{254,807,963},{254,863,907},{254,867,903},{257,803,964},{257,804,963},{257,863,904},{257,864,903},{263,804,957},{263,807,954},{263,854,907},{263,857,904},{264,803,957},{264,807,953},{264,853,907},{264,857,903},{267,803,954},{267,804,953},{267,853,904},{267,854,903},{273,805,946},{273,806,945},{273,845,906},{273,846,905},{275,803,946},{275,806,943},{275,843,906},{275,846,903},{276,803,945},{276,805,943},{276,843,905},{276,845,903},{302,754,968},{302,758,964},{302,764,958},{302,768,954},{304,752,968},{304,758,962},{304,762,958},{304,768,952},{308,752,964},{308,754,962},{308,762,954},{308,764,952},{320,746,958},{320,748,956},{320,756,948},{320,758,946},{326,740,958},{326,748,950},{326,750,948},{326,758,940},{328,740,956},{328,746,950},{328,750,946},{328,756,940},{340,726,958},{340,728,956},{340,756,928},{340,758,926},{346,720,958},{346,728,950},{346,750,928},{346,758,920},{348,720,956},{348,726,950},{348,750,926},{348,756,920},{350,726,948},{350,728,946},{350,746,928},{350,748,926},{352,704,968},{352,708,964},{352,764,908},{352,768,904},{354,702,968},{354,708,962},{354,762,908},{354,768,902},{356,720,948},{356,728,940},{356,740,928},{356,748,920},{358,702,964},{358,704,962},{358,720,946},{358,726,940},{358,740,926},{358,746,920},{358,762,904},{358,764,902},{362,704,958},{362,708,954},{362,754,908},{362,758,904},{364,702,958},{364,708,952},{364,752,908},{364,758,902},{368,702,954},{368,704,952},{368,752,904},{368,754,902},{402,635,987},{402,637,985},{402,685,937},{402,687,935},{402,753,869},{402,759,863},{402,763,859},{402,769,853},{403,725,896},{403,726,895},{403,752,869},{403,759,862},{403,762,859},{403,769,852},{403,795,826},{403,796,825},{405,632,987},{405,637,982},{405,682,937},{405,687,932},{405,723,896},{405,726,893},{405,793,826},{405,796,823},{406,723,895},{406,725,893},{406,793,825},{406,795,823},{407,632,985},{407,635,982},{407,682,935},{407,685,932},{409,752,863},{409,753,862},{409,762,853},{409,763,852},{420,735,869},{420,739,865},{420,765,839},{420,769,835},{423,705,896},{423,706,895},{423,795,806},{423,796,805},{425,703,896},{425,706,893},{425,730,869},{425,739,860},{425,760,839},{425,769,830},{425,793,806},{425,796,803},{426,703,895},{426,705,893},{426,793,805},{426,795,803},{429,730,865},{429,735,860},{429,760,835},{429,765,830},{430,725,869},{430,729,865},{430,765,829},{430,769,825},{432,605,987},{432,607,985},{432,685,907},{432,687,905},{435,602,987},{435,607,982},{435,682,907},{435,687,902},{435,720,869},{435,729,860},{435,760,829},{435,769,820},{437,602,985},{437,605,982},{437,682,905},{437,685,902},{439,720,865},{439,725,860},{439,760,825},{439,765,820},{452,703,869},{452,709,863},{452,763,809},{452,769,803},{453,702,869},{453,709,862},{453,762,809},{453,769,802},{459,702,863},{459,703,862},{459,762,803},{459,763,802},{460,725,839},{460,729,835},{460,735,829},{460,739,825},{462,703,859},{462,709,853},{462,753,809},{462,759,803},{463,702,859},{463,709,852},{463,752,809},{463,759,802},{465,720,839},{465,729,830},{465,730,829},{465,739,820},{469,702,853},{469,703,852},{469,720,835},{469,725,830},{469,730,825},{469,735,820},{469,752,803},{469,753,802},{482,605,937},{482,607,935},{482,635,907},{482,637,905},{485,602,937},{485,607,932},{485,632,907},{485,637,902},{487,602,935},{487,605,932},{487,632,905},{487,635,902},{493,705,826},{493,706,825},{493,725,806},{493,726,805},{495,703,826},{495,706,823},{495,723,806},{495,726,803},{496,703,825},{496,705,823},{496,723,805},{496,725,803},{502,643,879},{502,649,873},{502,673,849},{502,679,843},{503,624,897},{503,627,894},{503,642,879},{503,649,872},{503,672,849},{503,679,842},{503,694,827},{503,697,824},{504,623,897},{504,627,893},{504,693,827},{504,697,823},{507,623,894},{507,624,893},{507,693,824},{507,694,823},{509,642,873},{509,643,872},{509,672,843},{509,673,842},{523,604,897},{523,607,894},{523,694,807},{523,697,804},{524,603,897},{524,607,893},{524,693,807},{524,697,803},{527,603,894},{527,604,893},{527,693,804},{527,694,803},{542,603,879},{542,609,873},{542,673,809},{542,679,803},{543,602,879},{543,609,872},{543,672,809},{543,679,802},{549,602,873},{549,603,872},{549,672,803},{549,673,802},{572,603,849},{572,609,843},{572,643,809},{572,649,803},{573,602,849},{573,609,842},{573,642,809},{573,649,802},{579,602,843},{579,603,842},{579,642,803},{579,643,802},{593,604,827},{593,607,824},{593,624,807},{593,627,804},{594,603,827},{594,607,823},{594,623,807},{594,627,803},{597,603,824},{597,604,823},{597,623,804},{597,624,803}}

wlc1

求：3, 4, 19, 80, 339, 1436, 6083, 25768, ...... 的通解公式，

Treenewbee

wlc1 发表于 2024-2-9 12:13
求：3, 4, 19, 80, 339, 1436, 6083, 25768, ...... 的通解公式，

\[a_n=\left(\left(\frac{19}{10} \sqrt{5}-4\right) \left(\sqrt{5}+2\right)^n-\left(\frac{19}{10} \sqrt{5}+4\right) \left(2-\sqrt{5}\right)^n\right)\]

cz1

求：x^199+y^193+z^2726898=w^71,

求：x^(199*71*193)+y^(193*71*199)+z^2726898=w^(71*193*199),

求：x^2726897+y^2726897+z^2726898=w^2726897 ,

鲁老师的天文数字的答案，从地球到银河系的距离！

解：\(((2^{2726898}-2)^{13703})^{199}+((2^{2726898}-2)^{13987})^{193}+(2^{2726898}-2)^{2726898}=((2(2^{2726898}-2))^{38021})^{71}\) .

cz1

方程1：a^35+b^143=c^323 ，显然有解！

方程1各指数同时乘以2，

方程2：a^70+b^286=c^646 ，是否有解？

Treenewbee

cz1 发表于 2024-2-9 22:17
方程1：a^35+b^143=c^323 ，显然有解！

方程1各指数同时乘以2，

\[\left(3^{40375} 4^{21115} 5^{30888}\right)^{70}+\left(3^{9882} 4^{5168} 5^{7560}\right)^{286}=\left(3^{4375} 4^{2288} 5^{3347}\right)^{646}\]

lusishun

cz1 发表于 2024-2-9 00:36
请鲁老师试一试，

求：\(a^{27}+b^{28}=z^{32}\)

利用：X^14337+y^14336=z^14336
原方程的解：
a=(n^14336-1)^531,
b=（n^14336-1)^512,
c=[n(n^14886-1)]^448.

cz1

求：\(a^{27}+b^{28}=z^{32}\)

谢谢鲁思顺老师的解答，

利用：\(x^{14337}+y^{14336}=z^{14336}\)

原方程的解：\(((2^{14336}-1)^{531})^{27}+((2^{14336}-1)^{512})^{28}+((2(2^{14886}-1))^{448})^{32}\)

cz1

求：\(a^4+b^5=c^4\)

求：\(a^4+b^3=c^4\)