本原勾股数接龙

nyy

比如{3,4,5}（从小到大排列）
从3开头，5结尾，
下一个是
{5,12,13}
再下一个是
{13,84,85}
然后是
{85,132,157}
能一直接下去吗？
下一组勾股数是？

nyy

下一个是157, 12324,  12325

northwolves

{{3,4,5},{5,12,13},{13,84,85},{85,132,157},{157,12324,12325},{12325,6900,14125},{14125,1428,14197},{14197,100777404,100777405},{100777405,203121707170668,203121707170693},{203121707170693,95616165526116924,95616381276148045},{95616381276148045,182849847366914288850160655146428,182849847366914288850160655146453},{182849847366914288850160655146453,13850645550156987067060471532793450625941345396,13850645550156987067060471534000400387273043805},{13850645550156987067060471534000400387273043805,65428305388568517102261001305006125585041514172,66878273962295614845236375099451962164724439397}}

nyy

northwolves 发表于 2023-12-10 12:19
{{3,4,5},{5,12,13},{13,84,85},{85,132,157},{157,12324,12325},{12325,6900,14125},{14125,1428,14197},{ ...

试试20,21,29接龙

northwolves

nyy 发表于 2023-12-10 12:26
试试20,21,29接龙

1. f[c_List]:=Block[{d=Divisors[Last@c]},y=d [[Length[d]/2]];x=Last@c/y;{Last@c,(x^2-y^2)/2,(x^2+y^2)/2}];
   
2. NestList[f[#]&,{20,21,29},12]

复制代码

{{20,21,29},{29,420,421},{421,88620,88621},{88621,55980,104821},{104821,4012260,4013629},{4013629,9577417980,9577418821},{9577418821,345464309580,345597043021},{345597043021,89431350015540,89432017772821},{89432017772821,80773550174940,120509137457629},{120509137457629,24860923673242860,24861215745393829},{24861215745393829,23582437124790660,34266767998802029},{34266767998802029,349008770338204946990281620,349008770338204946991963829},{349008770338204946991963829,45651084143448622043727900,351981737106427315045608829}}

northwolves

或者直接计算亦可：

1. NestList[Values@Solve[x==#[[-1]]&&x^2+y^2==z^2&&GCD[x,y]==1&&z>y>0,{x,y,z},Integers][[1]]&,{3,4,5},12]

复制代码

{{3,4,5},{5,12,13},{13,84,85},{85,132,157},{157,12324,12325},{12325,89892,90733},{90733,2447844,2449525},{2449525,28350372,28455997},{28455997,295742791596,295742792965},{295742792965,171480834409712412,171480834409967437},{171480834409967437,633511848768467916,656310093705697045},{656310093705697045,1616599508725767821225590810932,1616599508725767821225590944157},{1616599508725767821225590944157,4158520496012961741299012805876,4461691012090851100342993272805}}

nyy

northwolves 发表于 2023-12-10 17:15
或者直接计算亦可：

1. x=12325;
   
2. Solve[x^2+y^2==z^2&&GCD[x,y]==1&&0<y<z,{y,z},Integers]
   

复制代码

求解结果
{{y -> 89892, z -> 90733},
{y -> 121212, z -> 121837},
{y -> 262668, z -> 262957},
{y -> 75952812, z -> 75952813}}
单单这个就有四个解

nyy

northwolves 发表于 2023-12-10 17:15
或者直接计算亦可：

帮你把结果美化一下：

\[\begin{array}{rrr}
3 & 4 & 5 \\
5 & 12 & 13 \\
13 & 84 & 85 \\
85 & 132 & 157 \\
157 & 12324 & 12325 \\
12325 & 89892 & 90733 \\
90733 & 2447844 & 2449525 \\
2449525 & 28350372 & 28455997 \\
28455997 & 295742791596 & 295742792965 \\
295742792965 & 171480834409712412 & 171480834409967437 \\
171480834409967437 & 633511848768467916 & 656310093705697045 \\
656310093705697045 & 1616599508725767821225590810932 & 1616599508725767821225590944157 \\
1616599508725767821225590944157 & 4158520496012961741299012805876 & 4461691012090851100342993272805 \\
\end{array}\]

nyy

northwolves 发表于 2023-12-10 13:03
{{20,21,29},{29,420,421},{421,88620,88621},{88621,55980,104821},{104821,4012260,4013629},{401362 ...

如果接龙的时候，要求里面至少一个素数。比如你的
{88621, 55980, 104821}, {104821, 4012260, 4013629}, {4013629, 9577417980, 9577418821}
这个里面都不是素数！
而
{421, 88620, 88621}里面，第一个421是素数

yuange1975

老是咋咋呼呼的，我还以为多牛呢。

不会连勾股数的通用公式都不知道吧？