一个完全由 0 和 1 组成的数是否有可能为完全平方数？

uk702

问题1: 10^m + 1不能为完全平方数。
证：易知 10^m + 1 是一个  3k - 1 型的数，固知非 3 的倍数若为完全平方数，必为  3k + 1 型，所以   10^m + 1  不能为完全平方数。

问题2： 10^m + 10^n + 1  不能为完全平方数。
证：易知  10^m + 10^n + 1   是   9k + 3 型，是 3 的倍数但不是 9 的倍数，所以不可能是完全平方数。

问题3： 10^m + 10^n + 10^p + 1 不能为完全平方数。
证：若 n=p，显然 10^m + 10^n + 10^p + 1 是可以为完全平方数，下面讨论仅限于 m>n>p>0 的情况。
（... 以下不会了 ...）

问题：一个完全由 0 和两个以的 1 组成的数是否有可能为完全平方数？ 若有，其最小值是 （  ） ？

northwolves

一个n位数只有1和0的概率是$\frac{1}{5^n}$,平方数的几率$\frac{1}{\sqrt{10}^n}$。假设两个事件独立，两个条件对同一个数成立的概率为$\frac{1}{(5\sqrt{10})^n}<\frac{1}{15^n}<\frac{1}{10^n}$。

northwolves

仅由两个非0不同数字组成的平方数：
{16, 25, 36, 49, 64, 81, 121, 144, 225, 441, 484, 676, 1444, 7744, 11881, 29929, 44944, 55225, 69696, 9696996, 6661661161}

northwolves

我以前搜索到10^44，没有新值

northwolves

Sporadic solutions

三个数字的参考mathe版主大作

uk702

northwolves 发表于 2024-7-9 14:31
Sporadic solutions

三个数字的参考mathe版主大作

那么，这个问题（一个完全由 0 和两个以的 1 组成的数是否有可能为完全平方数？）是不是已经有无解的答案了？还是说尚未解决？

northwolves

无解。应该不太难证明，楼主试试看

lihpb00

应该无解

northwolves

若$N=1******01$是一个平方数,则$\sqrtN=[13]......[19]$，N里面1的数量$n=9k+{0,1,4,7}$

northwolves

发现其他进制下可以有很多，如：

1. Select[Range@1000000, Max@IntegerDigits[#^2, 8] == 1 &]

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八进制： {1,3,8,24,64,192,512,1536,4096,11677,12288,32768,93416,98304,262144,747328,786432}

1. Select[Range@1000000, Max@IntegerDigits[#^2, 5] == 1 &]

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五进制： {1,5,25,125,625,3125,15625,78125,390625,972799}

1. Select[Range@1000000, Max@IntegerDigits[#^2, 7] == 1 &]

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七进制： {1,7,20,49,140,343,980,2401,6860,16807,48020,117649,336140,823543}

1. Select[Range@1000000, Max@IntegerDigits[#^2, 3] == 1 &]

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三进制： {1,2,3,6,9,11,16,18,19,27,29,33,48,54,55,57,81,83,87,99,143,144,162,163,165,171,243,245,249,261,262,297,421,429,432,451,486,487,489,495,513,729,731,735,747,783,786,889,891,1263,1287,1296,1331,1342,1353,1458,1459,1461,1467,1485,1487,1539,2187,2189,2193,2205,2241,2242,2323,2349,2358,2537,2573,2644,2662,2667,2673,3788,3789,3861,3888,3993,4026,4059,4374,4375,4377,4383,4401,4455,4461,4562,4617,6561,6563,6567,6579,6615,6688,6723,6726,6967,6969,7036,7047,7074,7082,7611,7696,7719,7932,7986,8001,8019,8035,11364,11367,11583,11645,11664,11675,11979,11987,12078,12177,12202,13122,13123,13125,13131,13149,13203,13205,13310,13365,13383,13660,13678,13686,13835,13851,19683,19685,19689,19701,19737,19845,19846,20064,20089,20169,20178,20762,20791,20869,20901,20907,21108,21130,21141,21222,21246,22833,23088,23157,23671,23796,23803,23958,23974,24003,24057,24073,24105,34092,34101,34127,34749,34935,34992,35003,35025,35937,35961,36234,36531,36606,39366,39367,39369,39375,39393,39447,39609,39615,39930,40095,40149,40980,41032,41034,41058,41505,41553,41669,59049,59051,59055,59067,59103,59211,59535,59538,60192,60265,60267,60496,60507,60534,60667,62286,62373,62607,62703,62721,63324,63390,63423,63666,63738,68499,69264,69471,69550,70972,71013,71388,71409,71874,71922,71983,72009,72171,72187,72219,72315,102276,102303,102381,104164,104247,104438,104805,104976,104987,105009,105075,107811,107883,108008,108413,108494,108702,109593,109818,110374,110402,118098,118099,118101,118107,118125,118179,118341,118343,118667,118827,118845,119747,119790,120285,120447,120530,122940,122942,123096,123102,123174,124515,124659,125007,177147,177149,177153,177165,177201,177309,177633,177634,178363,178605,178614,178778,180576,180795,180801,181488,181510,181521,181602,182001,186858,187119,187804,187821,188109,188163,188260,189818,189972,190170,190269,190324,190838,190998,191176,191214,204587,205497,205928,207792,208413,208650,208753,212916,213039,213325,214148,214164,214227,215622,215766,215927,215949,216027,216513,216529,216561,216657,216945,306828,306909,307143,312492,312741,313147,313314,314415,314928,314939,314961,315027,315225,323433,323649,324024,325239,325482,325513,326106,328779,329454,330623,331122,331206,354294,354295,354297,354303,354321,354375,354537,355023,355029,356001,356481,356535,359241,359370,360855,360884,360982,361043,361341,361590,368820,368826,369288,369306,369522,370883,373545,373715,373977,375021,375707,531441,531443,531447,531459,531495,531603,531927,532899,532902,533072,535087,535089,535815,535842,536285,536334,541676,541728,542385,542403,542782,544464,544530,544563,544806,545605,546003,560574,561357,561625,563412,563463,564327,564362,564489,564780,569454,569916,570510,570807,570862,570972,572514,572994,573528,573642,613761,616491,617784,623376,623483,625239,625950,626259,638748,639117,639656,639739,639975,641843,642444,642492,642681,642790,646866,646895,647297,647298,647781,647847,648081,649539,649555,649587,649683,649971,650835,920484,920727,921429,928747,937476,938223,939441,939942,943013,943166,943245,944784,944795,944817,944883,945081,945673,945675,970299,970947,972072,975717,976446,976539,977644,978318,986337,988362,991826,991869,993366,993618,993950,994033}