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

王守恩

王守恩 发表于 2025-2-18 10:50
{x -> 1, y -> 1, x -> 2, y -> 2, x -> 3, y -> 3, x -> 11, y -> 11, x -> 12, y -> 21, x -> 13, y -> ...

只要 x ——1, 2, 3, 11, 12, 13, 22, 26, 33, 101, 102, 103, 111, 112, 113, 121, 122, 202, 212, 264, 307, 836, 1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103,

我们太超前了——OEIS连这串数还没有。

northwolves给出了通项公式！我的不行。谢谢 northwolves！！！

{1, 2, 3, 11, 12, 13, 22, 26, 33, 101, 102, 103, 111, 112, 113, 121, 122, 202, 212, 264, 307, 836, 1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1102, 1103, 1111, 1112, 1113, 1121, 1122, 1202, 1212, 2002, 2012,
2022, 2285, 2636, 3168, 10001, 10002, 10003, 10011, 10012, 10013, 10021, 10022, 10031, 10101, 10102, 10103, 10111, 10112, 10113, 10121, 10122, 10201, 10202, 10211, 10212, 10221, 11002, 11003, 11011, 11012,
11013, 11021, 11022, 11031, 11102, 11103, 11111, 11112, 11113, 11121, 11122, 11202, 11211, 12002, 12012, 12102, 12202, 20002, 20012, 20022, 20102, 20112, 20122, 20508, 22865, 24846, 30693, 100001, 100002,
100003, 100011, 100012, 100013, 100021, 100022, 100031, 100101, 100102, 100103, 100111, 100112, 100113, 100121, 100122, 100201, 100202, 100211, 100212, 100221, 100301, 100311, 101002, 101003, 101011,
101012, 101013, 101021, 101022, 101031, 101101, 101102, 101103, 101111, 101112, 101113, 101121, 101122, 101201, 101202, 101211, 101212, 101301, 102002, 102011, 102012, 102021, 102022, 102102, 102111,
102121, 110002, 110003, 110011, 110012, 110013, 110021, 110022, 110031, 110102, 110103, 110111, 110112, 110113, 110121, 110122, 110202, 110211, 110212, 110221, 110922, 111002, 111003, 111012, 111013,
111021, 111022, 111031, 111102, 111103, 111111, 111112, 111121, 111202, 111211, 112002, 112012, 112102, 120002, 120012, 120102, 120112, 121002, 121102, 122002, 200002, 200012, 200022, 200102, 200112,
200122, 200202, 200212, 201012, 201022, 202012, 303577, 798644}

1. Select[Range@900000, IntegerQ[Power[IntegerReverse[#^2], (2)^-1]] && IntegerReverse[#^2] >= #^2 &]

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蔡家雄

存在两个自守平方数，

设 \(a , b\) 都是 \(n\) 位数，

若 \(a^2\) 的最后\(n\)位数字是\(a , b^2\) 的最后\(n\)位数字是\(b\) ,

则 \(a+b = 10^n+1\) .

是否存在自守立方数？如果存在，有几个自守立方数？

蔡家雄

自守立方数，，，

设 \(c\) 是 \(n\) 位数，

若 \(c^3\) 的最后\(n\)位数字是\(c\) , 则 \(c\) 为 自守立方数，

是否存在自守立方数？如果存在，有几个自守立方数？

王守恩

蔡家雄 发表于 2025-2-19 08:59
自守立方数，，，

设 \(c\) 是 \(n\) 位数，

  平方数 = 2,     立方数 = 3,  ......     注意观察得数相同的 k。
k=2, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=3, {1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001},
k=4, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=5, {1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501},
k=6, {1, 5, 6, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96, 176, 201, 376, 401, 576, 601, 625, 776, 801, 976, 1376, 2001, 3376, 4001, 5376, 6001, 7376, 8001, 9376, 20001, 29376, 40001, 49376, 60001, 69376, 80001, 89376, 90625, 109376, 200001},
k=7, {1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001},
k=8, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=9, {1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501},
k=10, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=11, {1, 4, 5, 6, 9, 11, 16, 19, 21, 24, 25, 29, 31, 36, 39, 41, 44, 49, 51, 56, 59, 61, 64, 69, 71, 75, 76, 79, 81, 84, 89, 91, 96, 99, 101, 125, 149, 151, 176, 199, 201, 224, 249, 251, 299, 301, 349, 351, 375, 376, 399, 401, 424, 449, 451, 499, 501},
k=12, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=13, {1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501},
k=14, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=15, {1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001},
k=16, {1, 5, 6, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96, 176, 201, 376, 401, 576, 601, 625, 776, 801, 976, 1376, 2001, 3376, 4001, 5376, 6001, 7376, 8001, 9376, 20001, 29376, 40001, 49376, 60001, 69376, 80001, 89376, 90625, 109376, 200001},
k=17, {1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 25, 32, 43, 49, 51, 57, 68, 75, 76, 93, 99, 125, 193, 249, 251, 307, 375, 376, 432, 443, 499, 501, 557, 568, 624, 625, 693, 749, 751, 807, 875, 943, 999, 1249, 1251, 1693, 1875, 2057, 2499, 2501},
k=18, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},
k=19, {1, 4, 5, 6, 9, 24, 25, 49, 51, 75, 76, 99, 125, 249, 251, 375, 376, 499, 501, 624, 625, 749, 751, 875, 999, 1249, 3751, 4375, 4999, 5001, 5625, 6249, 8751, 9375, 9376, 9999, 18751, 31249, 40625, 49999, 50001}
k=18, {1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376},

王守恩

蔡家雄 发表于 2025-2-16 17:06
例：343，1331，1367631， 是回文立方数，

问：回文可逆立方数是有限个？还是无限个？

{1, 2, 7, 11, 101, 111, 1001, 1011, 1101, 10001, 10011, 10101, 11001, 11011, 100001, 100011, 100101, 100111, 101001, 101011, 101101, 110001, 110011, 110101, 111001, 1000001,
1000011, 1000101, 1000111, 1001001,  1001011, 1001101, 1010001, 1010011, 1011001, 1100001, 1100011, 1100101, 1101001, 1110001, 10000001,  10000011, 10000101, 10000111,
10001001, 10001011, 10001101, 10010001, 10010011, 10010101, 10011001, 10100001,10100011, 10100101,10101001, 10110001, 11000001,11000011,11000101,11001001,11010001,
11100001, 100000001, 100000011, 100000101, 100000111, 100001001, 100001011, 100001101, 100010001, 100010011, 100011001, 100100001, 100100011, 100100101,  100101001,
100110001, 101000001, 101000011, 101000101, 101001001, 101100001, 110000001, 110000011, 110000101, 110001001, 110010001, 110100001, 111000001}

1. Select[FromDigits@IntegerDigits[#, 2] & /@ Range[2^19], IntegerQ[Power[IntegerReverse@(#^3), (3)^-1]] && Mod[#, 10] > 0 &]

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northwolves 的通项公式！ 要多少有多少！！ 谢谢 northwolves ！！！

ysr

蔡家雄 发表于 2025-2-27 01:32
求 \(x^2 - 2y^2 = 89\) 的正整数解

程序结果：
x=11  y=4
x=17  y=10
x=49  y=34
x=91  y=64
x=283  y=200
x=529  y=374
解可能是无穷的

ysr

蔡家雄 发表于 2025-2-27 01:32
求 \(x^2 - 2y^2 = 89\) 的正整数解

程序结果：
x=3  y=7
x=19  y=15
x=37  y=27
x=117  y=83
x=219  y=155
x=683  y=483

解可能是无穷的

ysr

蔡家雄 发表于 2025-2-27 04:36
求 \(x^2 - 2y^2 = 23\) 的正整数解

程序结果：
x=5  y=1
x=11  y=7
x=19  y=13
x=61  y=43
x=109  y=77
x=355  y=251
x=635  y=449

ysr

蔡家雄 发表于 2025-2-27 04:36
求 \(x^2 - 2y^2 = - 23\) 的正整数解

程序结果：
x=3  y=4
x=7  y=6
x=25  y=18
x=45  y=32
x=147  y=104
x=263  y=186
x=857  y=606

ysr

蔡家雄 发表于 2025-2-27 09:27
求 \(x^2 - 2y^2 = 46\) 的正整数解

程序结果：
x=8  y=3
x=12  y=7
x=36  y=25
x=64  y=45
x=208  y=147
x=372  y=263