Question Number 176101 by BaliramKumar last updated on 12/Sep/22
![x^2 − 4y^2 = 3z^2 [ x, y, z ∈ N ] HCF(x, y, z) = 1 like (4, 1, 2) x, y, z < 100 solve by computer programing](https://www.tinkutara.com/question/Q176101.png)
$${x}^{\mathrm{2}} \:−\:\mathrm{4}{y}^{\mathrm{2}} \:=\:\mathrm{3}{z}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\left[\:{x},\:{y},\:{z}\:\in\:\mathbb{N}\:\right]\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{HCF}\left({x},\:{y},\:{z}\right)\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{like}\:\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x},\:{y},\:{z}\:<\:\mathrm{100} \\ $$$${solve}\:{by}\:{computer}\:{programing} \\ $$
Commented by Ar Brandon last updated on 12/Sep/22
#include <stdio.h>
int main(void)
{
for (short x = 1; x < 100; x++)
for (short y = 1; y < 100; y++)
for (short z = 1; z < 100; z++)
if (x*x - 4*y*y == 3*z*z)
printf("(x,y,z)=(%hd,%hd,%hd)\n",x,y,z);
return 0;
}
Commented by Ar Brandon last updated on 12/Sep/22
$$\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right),\:\left(\mathrm{14},\:\mathrm{1},\:\mathrm{8}\right),\:\left(\mathrm{26},\:\mathrm{11},\:\mathrm{8}\right),\:\left(\mathrm{28},\:\mathrm{11},\:\mathrm{10}\right), \\ $$$$\left(\mathrm{28},\:\mathrm{13},\:\mathrm{6}\right),\:\left(\mathrm{38},\:\mathrm{13},\:\mathrm{16}\right),\:\left(\mathrm{52},\:\mathrm{1},\:\mathrm{30}\right),\:\left(\mathrm{52},\:\mathrm{23},\:\mathrm{14}\right), \\ $$$$\left(\mathrm{62},\:\mathrm{23},\:\mathrm{24}\right),\:\left(\mathrm{74},\:\mathrm{13},\:\mathrm{40}\right),\:\left(\mathrm{76},\:\mathrm{11},\:\mathrm{42}\right),\:\left(\mathrm{76},\:\mathrm{37},\:\mathrm{10}\right), \\ $$$$\left(\mathrm{86},\:\mathrm{11},\:\mathrm{48}\right),\:\left(\mathrm{98},\:\mathrm{47},\:\mathrm{16}\right). \\ $$
Commented by Ar Brandon last updated on 12/Sep/22
#include <iostream>
using namespace std;
int main(void)
{
for (short x = 1; x < 100; x++)
for (short y = 1; y < 100; y++)
for (short z = 1; z < 100; z++)
if (x*x - 4*y*y == 3*z*z)
cout<<"(x,y,z)=("<<x<<","<<y<<","<<z<<")"<<endl;
return 0;
}
Commented by BaliramKumar last updated on 13/Sep/22
$${thank}\:{you}\:{sir}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${how}\:{taken}\:{time}\:{in}\:{programing} \\ $$$${see}\:\downarrow\downdownarrows\downdownarrows \\ $$
Commented by BaliramKumar last updated on 12/Sep/22
Commented by BaliramKumar last updated on 13/Sep/22
$${A}\:{tringle}'{s}\:\:{side}\:{in}\:{A}.{P}.\:\&\:{area}\:{is}\:{rational} \\ $$$$\left({a},\:{b},\:{c}\right)\:=\:\left(\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right),\:\:\left(\mathrm{13},\:\mathrm{14},\:\mathrm{15}\right),\:\:\left(\mathrm{15},\:\mathrm{26},\:\mathrm{37}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{17},\:\mathrm{28},\:\mathrm{39}\right),\:\left(\mathrm{15},\:\mathrm{28},\:\mathrm{41}\right),\:\left(\mathrm{25},\:\mathrm{38},\:\mathrm{51}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{51},\:\mathrm{52},\:\mathrm{53}\right),\:\left(\mathrm{29},\:\mathrm{52},\:\mathrm{75}\right),\:\left(\mathrm{39},\:\mathrm{62},\:\mathrm{85}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………….. \\ $$