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勾股数新公式

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发表于 2019-11-20 18:32 | 显示全部楼层
蔡家雄 发表于 2019-11-20 13:11
等周长本原三角形举例:
a=m^2-n^2,    b=2mn,     c=m^2+n^2

蔡家雄!还得来几个,我没找到规律。等周长本原三角形举例:
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发表于 2019-11-21 16:46 | 显示全部楼层
蔡家雄 发表于 2019-11-20 13:11
等周长本原三角形举例:
a=m^2-n^2,    b=2mn,     c=m^2+n^2

等周长本原三角形举例

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发表于 2019-11-21 16:52 | 显示全部楼层
蔡家雄 发表于 2019-11-20 13:11
等周长本原三角形举例:
a=m^2-n^2,    b=2mn,     c=m^2+n^2


4060420^2+109478589^2=109553861^2,4060420+109478589+109553861=223092870
15127629^2+103432420^2=104532821^2,15127629+103432420+104532821=223092870
28377492^2+ 95289845^2=99425533^2,  28377492+95289845+99425533=223092870
35813645^2+90215268^2=97063957^2,   35813645+90215268+97063957=223092870
38540645^2+88251828^2=96300397^2,   38540645+88251828+96300397=223092870
54658212^2+75348845^2=93085813^2,   54658212+75348845+93085813=223092870
56605461^2+73620820^2=92866589^2,   56605461+73620820+92866589=223092870
64408461^2+66270820^2=92413589^2,   64408461+66270820+92413589=223092870
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发表于 2019-12-1 15:46 | 显示全部楼层
蔡家雄 发表于 2019-12-1 15:05
面积为100的整数边三角形有多少个?

这个自己编程去算好了。
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发表于 2019-12-2 06:47 | 显示全部楼层
本帖最后由 王守恩 于 2019-12-2 07:09 编辑
wlc1 发表于 2019-12-2 05:58
图老师能找到面积为7的整边三角形吗?

图老师能找到面积为8的整边三角形吗?



a,b,c是三角形三边(整数),下面的公式可以有三角形全部面积(整数)解。

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好好好  发表于 2020-7-10 18:20
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发表于 2019-12-2 16:15 | 显示全部楼层
王守恩 发表于 2019-12-2 06:47
a,b,c是三角形三边(整数),下面的公式可以有三角形全部面积(整数)解。

a,b,c是三角形三边(整数),下面的数组都可以有面积(整数)解。
{a -> 03, b -> 04, c -> 05}, {a -> 03, b -> 25, c -> 26},
{a -> 04, b -> 13, c -> 15}, {a -> 04, b -> 51, c -> 53},
{a -> 05, b -> 12, c -> 13}, {a -> 05, b -> 29, c -> 30},
{a -> 05, b -> 51, c -> 52}, {a -> 06, b -> 08, c -> 10},
{a -> 06, b -> 25, c -> 29}, {a -> 06, b -> 50, c -> 52},
{a -> 07, b -> 15, c -> 20}, {a -> 07, b -> 24, c -> 25},
{a -> 07, b -> 65, c -> 68}, {a -> 08, b -> 15, c -> 17},
{a -> 08, b -> 26, c -> 30}, {a -> 08, b -> 29, c -> 35},
{a -> 09, b -> 10, c -> 17}, {a -> 09, b -> 12, c -> 15},
{a -> 09, b -> 40, c -> 41}, {a -> 09, b -> 65, c -> 70},
{a -> 09, b -> 73, c -> 80}, {a -> 09, b -> 75, c -> 78},
{a -> 10, b -> 17, c -> 21}, {a -> 10, b -> 24, c -> 26},
{a -> 10, b -> 35, c -> 39}, {a -> 10, b -> 58, c -> 60},
{a -> 11, b -> 13, c -> 20}, {a -> 11, b -> 25, c -> 30},
{a -> 11, b -> 60, c -> 61}, {a -> 12, b -> 16, c -> 20},
{a -> 12, b -> 17, c -> 25}, {a -> 12, b -> 35, c -> 37},
{a -> 12, b -> 39, c -> 45}, {a -> 12, b -> 50, c -> 58},
{a -> 12, b -> 55, c -> 65}, {a -> 13, b -> 14, c -> 15},
{a -> 13, b -> 20, c -> 21}, {a -> 13, b -> 30, c -> 37},
{a -> 13, b -> 37, c -> 40}, {a -> 13, b -> 40, c -> 45},
{a -> 13, b -> 40, c -> 51}, {a -> 13, b -> 68, c -> 75},
{a -> 13, b -> 84, c -> 85}, {a -> 14, b -> 30, c -> 40},
{a -> 14, b -> 48, c -> 50}, {a -> 14, b -> 61, c -> 65},
{a -> 15, b -> 20, c -> 25}, {a -> 15, b -> 26, c -> 37},
{a -> 15, b -> 28, c -> 41}, {a -> 15, b -> 34, c -> 35},
{a -> 15, b -> 36, c -> 39}, {a -> 15, b -> 37, c -> 44},
{a -> 15, b -> 41, c -> 52}, {a -> 15, b -> 52, c -> 61},
{a -> 16, b -> 25, c -> 39}, {a -> 16, b -> 30, c -> 34},
{a -> 16, b -> 52, c -> 60}, {a -> 16, b -> 58, c -> 70},
{a -> 16, b -> 63, c -> 65}, {a -> 17, b -> 25, c -> 26},
{a -> 17, b -> 25, c -> 28}, {a -> 17, b -> 28, c -> 39},
{a -> 17, b -> 39, c -> 44}, {a -> 17, b -> 40, c -> 41},
{a -> 17, b -> 55, c -> 60}, {a -> 17, b -> 65, c -> 80},
{a -> 18, b -> 20, c -> 34}, {a -> 18, b -> 24, c -> 30},
{a -> 18, b -> 75, c -> 87}, {a -> 18, b -> 80, c -> 82},
{a -> 19, b -> 20, c -> 37}, {a -> 19, b -> 60, c -> 73},
{a -> 20, b -> 21, c -> 29}, {a -> 20, b -> 34, c -> 42},
{a -> 20, b -> 37, c -> 51}, {a -> 20, b -> 48, c -> 52},
{a -> 20, b -> 51, c -> 65}, {a -> 20, b -> 53, c -> 55},
{a -> 20, b -> 65, c -> 75}, {a -> 20, b -> 70, c -> 78},
{a -> 21, b -> 28, c -> 35}, {a -> 21, b -> 41, c -> 50},
{a -> 21, b -> 45, c -> 60}, {a -> 21, b -> 61, c -> 68},
{a -> 21, b -> 72, c -> 75}, {a -> 22, b -> 26, c -> 40},
{a -> 22, b -> 50, c -> 60}, {a -> 24, b -> 32, c -> 40},
{a -> 24, b -> 34, c -> 50}, {a -> 24, b -> 35, c -> 53},
{a -> 24, b -> 45, c -> 51}, {a -> 24, b -> 70, c -> 74},
{a -> 25, b -> 29, c -> 36}, {a -> 25, b -> 33, c -> 52},
{a -> 25, b -> 34, c -> 39}, {a -> 25, b -> 38, c -> 51},
{a -> 25, b -> 39, c -> 40}, {a -> 25, b -> 39, c -> 56},
{a -> 25, b -> 51, c -> 52}, {a -> 25, b -> 51, c -> 74},
{a -> 25, b -> 52, c -> 63}, {a -> 25, b -> 60, c -> 65},
{a -> 25, b -> 63, c -> 74}, {a -> 25, b -> 74, c -> 77},
{a -> 26, b -> 28, c -> 30}, {a -> 26, b -> 35, c -> 51},
{a -> 26, b -> 40, c -> 42}, {a -> 26, b -> 51, c -> 55},
{a -> 26, b -> 51, c -> 73}, {a -> 26, b -> 60, c -> 74},
{a -> 26, b -> 73, c -> 97}, {a -> 27, b -> 29, c -> 52},
{a -> 27, b -> 30, c -> 51}, {a -> 27, b -> 36, c -> 45},
{a -> 28, b -> 45, c -> 53}, {a -> 28, b -> 60, c -> 80},
{a -> 28, b -> 65, c -> 89}, {a -> 29, b -> 35, c -> 48},
{a -> 29, b -> 52, c -> 69}, {a -> 29, b -> 52, c -> 75},
{a -> 29, b -> 60, c -> 85}, {a -> 29, b -> 65, c -> 68},
{a -> 30, b -> 40, c -> 50}, {a -> 30, b -> 51, c -> 63},
{a -> 30, b -> 52, c -> 74}, {a -> 30, b -> 56, c -> 82},
{a -> 30, b -> 68, c -> 70}, {a -> 31, b -> 68, c -> 87},
{a -> 32, b -> 50, c -> 78}, {a -> 32, b -> 53, c -> 75},
{a -> 32, b -> 60, c -> 68}, {a -> 33, b -> 34, c -> 65},
{a -> 33, b -> 39, c -> 60}, {a -> 33, b -> 41, c -> 58},
{a -> 33, b -> 44, c -> 55}, {a -> 33, b -> 56, c -> 65},
{a -> 33, b -> 58, c -> 85}, {a -> 34, b -> 50, c -> 52},
{a -> 34, b -> 50, c -> 56}, {a -> 34, b -> 55, c -> 87},
{a -> 34, b -> 56, c -> 78}, {a -> 34, b -> 61, c -> 75},
{a -> 34, b -> 65, c -> 93}, {a -> 35, b -> 44, c -> 75},
{a -> 35, b -> 52, c -> 73}, {a -> 35, b -> 53, c -> 66},
{a -> 36, b -> 40, c -> 68}, {a -> 36, b -> 48, c -> 60},
{a -> 36, b -> 51, c -> 75}, {a -> 36, b -> 61, c -> 65},
{a -> 37, b -> 39, c -> 52}, {a -> 38, b -> 40, c -> 74},
{a -> 39, b -> 41, c -> 50}, {a -> 39, b -> 42, c -> 45},
{a -> 39, b -> 52, c -> 65}, {a -> 39, b -> 55, c -> 82},
{a -> 39, b -> 58, c -> 95}, {a -> 39, b -> 60, c -> 63},
{a -> 40, b -> 42, c -> 58}, {a -> 40, b -> 51, c -> 77},
{a -> 41, b -> 50, c -> 73}, {a -> 41, b -> 50  c -> 89},
{a -> 41, b -> 51, c -> 58}, {a -> 42, b -> 56, c -> 70},
{a -> 44, b -> 52, c -> 80}, {a -> 45, b -> 50, c -> 85},
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 楼主| 发表于 2019-12-9 10:09 | 显示全部楼层
若n为正整数,

则面积为(6n)^2 的整边三角形(a, b, c)一定存在。

猜想:此时,这个整边三角形的周长(a+b+c)一定能被9整除。

若猜想为真,则三边长和面积(4个数)都是完全平方数的三角形不存在。

证明思想:类似于费尔马著名的“无限递降法”。

推论:存在无穷个k, 满足

面积为(6kn)^2且周长为完全平方数的整边三角形(ka, kb, kc)一定存在。

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 楼主| 发表于 2019-12-28 19:12 | 显示全部楼层
本帖最后由 蔡家雄 于 2020-2-14 04:06 编辑

设n为正整数,d为公差,

则 n*(n+d)*(n+2d)*(n+3d)+d^4 =(n^2+3nd+d^2)^2


公共弦勾股数的个数公式

它与公共弦c的4x-1 型素数 无关,

均与公共弦c的4x+1型素数 有关,

设公共弦c中有t个4x+1型的素数,

它的指数为r1, r2, ... , rt,

则公共弦勾股数的个数公式为

[(1+2r1)*(1+2r2)*...*(1+2rt) -1]/2


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 楼主| 发表于 2020-1-27 22:14 | 显示全部楼层
等周长本原三角形举例:

(153868,9435,154157)

(99660,86099,131701)

(43660,133419,140381)

(13260,151811,152389)

等周长本原三角形举例:
a=m^2-n^2,    b=2mn,     c=m^2+n^2

(m,n)=(60,59)=(68,37)=(84,1)

(m,n)=(187,8)=(165,56)=(143,112)

(m,n)=(195,14)=(165,82)=(143,142)

(m,n)=(170,103)=(182,73)=(210,11)

(m,n)=(182,103)=(190,83)=(210,37)

等周长本原三角形举例:
a=m^2-n^2,    b=2mn,     c=m^2+n^2

(m,n)=(390,307)=(410,253)=(442,173)=(510,23)

(m,n)=(374,367)=(418,245)=(442,185)=(494,67)

(m,n)=(390,341)=(430,233)=(442,203)=(510,49)

(m,n)=(385,356)=(399,316)=(429,236)=(455,172)

(m,n)=(490,479)=(510,421)=(570,263)=(646,89)

(m,n)=(506,469)=(550,347)=(598,227)=(650,109)

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 楼主| 发表于 2020-1-27 22:18 | 显示全部楼层
周长和面积相等且均为整数的三角形

{74,182,192},{84,164,200},{96,149,203},{104,140,204}

周长448, 面积是6720.

周长和面积相等且均为整数的三角形

有五个的,周长是7546,面积是2522520:

(1901,2772,2873)  (1914,27232909)  (1925,2693,2928)  (2018,2525,3003)  (2213,2288,3045)

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