勾股数新公式

Treenewbee

1. a = 23; Table[{m, Table[{k, f[10^m, a, k]}, {k, {158, 159}}]}, {m,
   
2.   100}]

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{{1,{{158,0},{159,0}}},{2,{{158,0},{159,0}}},{3,{{158,0},{159,0}}},{4,{{158,0},{159,-1}}},{5,{{158,0},{159,-1}}},{6,{{158,0},{159,-1}}},{7,{{158,0},{159,-1}}},{8,{{158,0},{159,-1}}},{9,{{158,0},{159,-1}}},{10,{{158,0},{159,-1}}},{11,{{158,0},{159,-1}}},{12,{{158,0},{159,-1}}},{13,{{158,0},{159,-1}}},{14,{{158,0},{159,-1}}},{15,{{158,0},{159,-1}}},{16,{{158,0},{159,-1}}},{17,{{158,0},{159,-1}}},{18,{{158,0},{159,-1}}},{19,{{158,0},{159,-1}}},{20,{{158,0},{159,-1}}},{21,{{158,0},{159,-1}}},{22,{{158,0},{159,-1}}},{23,{{158,0},{159,-1}}},{24,{{158,0},{159,-1}}},{25,{{158,0},{159,-1}}},{26,{{158,0},{159,-1}}},{27,{{158,0},{159,-1}}},{28,{{158,0},{159,-1}}},{29,{{158,0},{159,-1}}},{30,{{158,0},{159,-1}}},{31,{{158,0},{159,-1}}},{32,{{158,0},{159,-1}}},{33,{{158,0},{159,-1}}},{34,{{158,0},{159,-1}}},{35,{{158,0},{159,-1}}},{36,{{158,0},{159,-1}}},{37,{{158,0},{159,-1}}},{38,{{158,0},{159,-1}}},{39,{{158,0},{159,-1}}},{40,{{158,0},{159,-1}}},{41,{{158,0},{159,-1}}},{42,{{158,0},{159,-1}}},{43,{{158,0},{159,-1}}},{44,{{158,0},{159,-1}}},{45,{{158,0},{159,-1}}},{46,{{158,0},{159,-1}}},{47,{{158,0},{159,-1}}},{48,{{158,0},{159,-1}}},{49,{{158,0},{159,-1}}},{50,{{158,0},{159,-1}}},{51,{{158,0},{159,-1}}},{52,{{158,0},{159,-1}}},{53,{{158,0},{159,-1}}},{54,{{158,0},{159,-1}}},{55,{{158,0},{159,-1}}},{56,{{158,0},{159,-1}}},{57,{{158,0},{159,-1}}},{58,{{158,0},{159,-1}}},{59,{{158,0},{159,-1}}},{60,{{158,0},{159,-1}}},{61,{{158,0},{159,-1}}},{62,{{158,0},{159,-1}}},{63,{{158,0},{159,-1}}},{64,{{158,0},{159,-1}}},{65,{{158,0},{159,-1}}},{66,{{158,0},{159,-1}}},{67,{{158,0},{159,-1}}},{68,{{158,0},{159,-1}}},{69,{{158,0},{159,-1}}},{70,{{158,0},{159,-1}}},{71,{{158,0},{159,-1}}},{72,{{158,0},{159,-1}}},{73,{{158,0},{159,-1}}},{74,{{158,0},{159,-1}}},{75,{{158,0},{159,-1}}},{76,{{158,0},{159,-1}}},{77,{{158,0},{159,-1}}},{78,{{158,0},{159,-1}}},{79,{{158,0},{159,-1}}},{80,{{158,0},{159,-1}}},{81,{{158,0},{159,-1}}},{82,{{158,0},{159,-1}}},{83,{{158,0},{159,-1}}},{84,{{158,0},{159,-1}}},{85,{{158,0},{159,-1}}},{86,{{158,0},{159,-1}}},{87,{{158,0},{159,-1}}},{88,{{158,0},{159,-1}}},{89,{{158,0},{159,-1}}},{90,{{158,0},{159,-1}}},{91,{{158,0},{159,-1}}},{92,{{158,0},{159,-1}}},{93,{{158,0},{159,-1}}},{94,{{158,0},{159,-1}}},{95,{{158,0},{159,-1}}},{96,{{158,0},{159,-1}}},{97,{{158,0},{159,-1}}},{98,{{158,0},{159,-1}}},{99,{{158,0},{159,-1}}},{100,{{158,0},{159,-1}}}}

如上，a=23,b=159在n=10^4 就不相等了，但b=158在n=10^100依然相等

王守恩

Treenewbee 发表于 2023-11-30 10:14
{{1,{{158,0},{159,0}}},{2,{{158,0},{159,0}}},{3,{{158,0},{159,0}}},{4,{{158,0},{159,-1}}},{5,{{1 ...

谢谢 Treenewbee！northwolves 给出了相同的答案。谢谢 northwolves！

1. Array[p = Log@Pi; {#, {Ceiling[#/p], Floor[#/(p - 1)]}} &, 100]

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{{1,{1,6}},{2,{2,13}},{3,{3,20}},{4,{4,27}},{5,{5,34}},{6,{6,41}},{7,{7,48}},{8,{7,55}},{9,{8,62}},{10,{9,69}},
{11,{10,76}},{12,{11,82}},{13,{12,89}},{14,{13,96}},{15,{14,103}},{16,{14,110}},{17,{15,117}},
{18,{16,124}},{19,{17,131}},{20,{18,138}},{21,{19,145}},{22,{20,152}},{23,{21,158}},{24,{21,165}},
{25,{22,172}},{26,{23,179}},{27,{24,186}},{28,{25,193}},{29,{26,200}},{30,{27,207}},{31,{28,214}},
{32,{28,221}},{33,{29,228}},{34,{30,234}},{35,{31,241}},{36,{32,248}},{37,{33,255}},{38,{34,262}},
{39,{35,269}},{40,{35,276}},{41,{36,283}},{42,{37,290}},{43,{38,297}},{44,{39,304}},{45,{40,310}},
{46,{41,317}},{47,{42,324}},{48,{42,331}},{49,{43,338}},{50,{44,345}},{51,{45,352}},{52,{46,359}},
{53,{47,366}},{54,{48,373}},{55,{49,380}},{56,{49,386}},{57,{50,393}},{58,{51,400}},{59,{52,407}},
{60,{53,414}},{61,{54,421}},{62,{55,428}},{63,{56,435}},{64,{56,442}},{65,{57,449}},{66,{58,456}},
{67,{59,462}},{68,{60,469}},{69,{61,476}},{70,{62,483}},{71,{63,490}},{72,{63,497}},{73,{64,504}},
{74,{65,511}},{75,{66,518}},{76,{67,525}},{77,{68,532}},{78,{69,538}},{79,{70,545}},{80,{70,552}},
{81,{71,559}},{82,{72,566}},{83,{73,573}},{84,{74,580}},{85,{75,587}},{86,{76,594}},{87,{77,601}},
{88,{77,608}},{89,{78,614}},{90,{79,621}},{91,{80,628}},{92,{81,635}},{93,{82,642}},{94,{83,649}},
{95,{83,656}},{96,{84,663}},{97,{85,670}},{98,{86,677}},{99,{87,684}},{100,{88,690}}}

王守恩

Treenewbee 发表于 2023-11-30 10:03
506楼很容易证明的，王老师您想复杂了

506楼(503楼)的思想还是领悟不了。EulerGamma=0.5772156649
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a/b}{\sqrt[n]{EulerGamma}}\bigg\rceil\)=n,  n=1,2,3,4,5,...,  这样的{a,b}是怎样的一些数对？
这些数据就已经错了？
a=01, b=？,
a=02, b=03,
a=03, b=04--5,
a=04, b=05--7,
a=05, b=06--9,
a=06, b=07--10,
a=07, b=08--12,
a=08, b=09--14,
a=09, b=10--16,
a=10, b=11--18,
a=11, b=12--20,
a=12, b=13--21,
后面的一串数: 1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 40, 41, 43,

1. Table[Floor[a/(-Log[EulerGamma])], {a, 1, 68}]

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前面的一串数怎么也出不来？要不这些数据就已经错了？

Treenewbee

王守恩 发表于 2023-12-1 10:14
506楼(503楼)的思想还是领悟不了。EulerGamma=0.5772156649
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a ...

1. Table[StringRiffle@{"a=",a,"\tb=",-Floor[a/(Log[EulerGamma]-1)],"-",-Ceiling[a/Log[EulerGamma]]},{a,50}]//MatrixForm

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a= 1         b= 1 - 1
a= 2         b= 2 - 3
a= 3         b= 2 - 5
a= 4         b= 3 - 7
a= 5         b= 4 - 9
a= 6         b= 4 - 10
a= 7         b= 5 - 12
a= 8         b= 6 - 14
a= 9         b= 6 - 16
a= 10         b= 7 - 18
a= 11         b= 8 - 20
a= 12         b= 8 - 21
a= 13         b= 9 - 23
a= 14         b= 10 - 25
a= 15         b= 10 - 27
a= 16         b= 11 - 29
a= 17         b= 11 - 30
a= 18         b= 12 - 32
a= 19         b= 13 - 34
a= 20         b= 13 - 36
a= 21         b= 14 - 38
a= 22         b= 15 - 40
a= 23         b= 15 - 41
a= 24         b= 16 - 43
a= 25         b= 17 - 45
a= 26         b= 17 - 47
a= 27         b= 18 - 49
a= 28         b= 19 - 50
a= 29         b= 19 - 52
a= 30         b= 20 - 54
a= 31         b= 21 - 56
a= 32         b= 21 - 58
a= 33         b= 22 - 60
a= 34         b= 22 - 61
a= 35         b= 23 - 63
a= 36         b= 24 - 65
a= 37         b= 24 - 67
a= 38         b= 25 - 69
a= 39         b= 26 - 70
a= 40         b= 26 - 72
a= 41         b= 27 - 74
a= 42         b= 28 - 76
a= 43         b= 28 - 78
a= 44         b= 29 - 80
a= 45         b= 30 - 81
a= 46         b= 30 - 83
a= 47         b= 31 - 85
a= 48         b= 31 - 87
a= 49         b= 32 - 89
a= 50         b= 33 - 90

Treenewbee

王守恩 发表于 2023-12-1 10:14
506楼(503楼)的思想还是领悟不了。EulerGamma=0.5772156649
a,b是正整数,  满足 \(\bigg\lceil\frac{n-a ...

1. Table[StringRiffle@{"a=",a,"\tb=",-Floor[a/(Log[EulerGamma]-1)],"-",-Ceiling[a/Log[EulerGamma]]},{a,50}]//MatrixForm

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a= 1         b= 1 - 1
a= 2         b= 2 - 3
a= 3         b= 2 - 5
a= 4         b= 3 - 7
a= 5         b= 4 - 9
a= 6         b= 4 - 10
a= 7         b= 5 - 12
a= 8         b= 6 - 14
a= 9         b= 6 - 16
a= 10         b= 7 - 18
a= 11         b= 8 - 20
a= 12         b= 8 - 21
a= 13         b= 9 - 23
a= 14         b= 10 - 25
a= 15         b= 10 - 27
a= 16         b= 11 - 29
a= 17         b= 11 - 30
a= 18         b= 12 - 32
a= 19         b= 13 - 34
a= 20         b= 13 - 36
a= 21         b= 14 - 38
a= 22         b= 15 - 40
a= 23         b= 15 - 41
a= 24         b= 16 - 43
a= 25         b= 17 - 45
a= 26         b= 17 - 47
a= 27         b= 18 - 49
a= 28         b= 19 - 50
a= 29         b= 19 - 52
a= 30         b= 20 - 54
a= 31         b= 21 - 56
a= 32         b= 21 - 58
a= 33         b= 22 - 60
a= 34         b= 22 - 61
a= 35         b= 23 - 63
a= 36         b= 24 - 65
a= 37         b= 24 - 67
a= 38         b= 25 - 69
a= 39         b= 26 - 70
a= 40         b= 26 - 72
a= 41         b= 27 - 74
a= 42         b= 28 - 76
a= 43         b= 28 - 78
a= 44         b= 29 - 80
a= 45         b= 30 - 81
a= 46         b= 30 - 83
a= 47         b= 31 - 85
a= 48         b= 31 - 87
a= 49         b= 32 - 89
a= 50         b= 33 - 90

Treenewbee

验证如下：

1. f[n_,a_,b_]:=Ceiling[(n-a/b)/EulerGamma^(1/n)]-n;
   
2. a=10;Table[{b,f[10^50,a,b]},{b,30}]
   
3. a=12;Table[{b,f[10^50,a,b]},{b,30}]

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{{1,-9},{2,-4},{3,-2},{4,-1},{5,-1},{6,-1},{7,0},{8,0},{9,0},{10,0},{11,0},{12,0},{13,0},{14,0},{15,0},{16,0},{17,0},{18,0},{19,1},{20,1},{21,1},{22,1},{23,1},{24,1},{25,1},{26,1},{27,1},{28,1},{29,1},{30,1}}

{{1,-11},{2,-5},{3,-3},{4,-2},{5,-1},{6,-1},{7,-1},{8,0},{9,0},{10,0},{11,0},{12,0},{13,0},{14,0},{15,0},{16,0},{17,0},{18,0},{19,0},{20,0},{21,0},{22,1},{23,1},{24,1},{25,1},{26,1},{27,1},{28,1},{29,1},{30,1}}

王守恩

Treenewbee 发表于 2023-12-1 12:10
a= 1         b= 1 - 1
a= 2         b= 2 - 3
a= 3         b= 2 - 5

譬如：a= 10     b= 7 - 10    n=1就不对了？   后面是对的。
Table[Ceiling[(n - a/b)/Power[EulerGamma, (n)^-1]], {a, 10, 10}, {b, 7, 12}, {n, 1, 5}]
{{0, 1, 2, 3, 4}, {0, 1, 3, 4, 5}, {0, 2, 3, 4, 5}, {0, 2, 3, 4, 5}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5}}}

Treenewbee

n=1时，EulerGamma^(1/n)太小，会引起较大的误差

王守恩

Treenewbee 发表于 2023-12-2 02:03
n=1时，EulerGamma^(1/n)太小，会引起较大的误差

n=1时，EulerGamma^(1/n)=EulerGamma，

1. Limit[Ceiling[(1 - a/(a + 1))/EulerGamma], a -> \[Infinity]]

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1

1. Array[{#, # + 1, Floor[-#/Log[EulerGamma]]} &, 30]

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{{1, 2, 1}, {2, 3, 3}, {3, 4, 5}, {4, 5, 7}, {5, 6, 9}, {6, 7, 10}, {7, 8, 12}, {8, 9, 14}, {9, 10, 16},
{10, 11, 18},{11, 12, 20}, {12, 13, 21}, {13, 14, 23}, {14, 15, 25}, {15, 16, 27}, {16, 17, 29},
{17, 18, 30}, {18, 19, 32}, {19, 20, 34}, {20, 21, 36}, {21, 22, 38}, {22, 23, 40}, {23, 24, 41},
{24, 25, 43}, {25, 26, 45}, {26, 27, 47}, {27, 28, 49}, {28, 29, 50}, {29, 30, 52}, {30, 31, 54},

蔡家雄

求解方程：\(y^2=8*x^2+1\) 的正整数解，