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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

$${x}^{\mathrm{2}} \:−\:\mathrm{4}{y}^{\mathrm{2}} \:=\:\mathrm{3}{z}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\left[\:{x},\:{y},\:{z}\:\in\:\mathbb{N}\:\right]\: \\ $$ $$\:\:\:\:\:\:\:\:\:\:{HCF}\left({x},\:{y},\:{z}\right)\:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{like}\:\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x},\:{y},\:{z}\:<\:\mathrm{100} \\ $$ $${solve}\:{by}\:{computer}\:{programing} \\ $$

Commented byAr 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 byAr Brandon last updated on 12/Sep/22

(4, 1, 2), (14, 1, 8), (26, 11, 8), (28, 11, 10), (28, 13, 6), (38, 13, 16), (52, 1, 30), (52, 23, 14), (62, 23, 24), (74, 13, 40), (76, 11, 42), (76, 37, 10), (86, 11, 48), (98, 47, 16).

$$\left(\mathrm{4},\:\mathrm{1},\:\mathrm{2}\right),\:\left(\mathrm{14},\:\mathrm{1},\:\mathrm{8}\right),\:\left(\mathrm{26},\:\mathrm{11},\:\mathrm{8}\right),\:\left(\mathrm{28},\:\mathrm{11},\:\mathrm{10}\right), \\ $$ $$\left(\mathrm{28},\:\mathrm{13},\:\mathrm{6}\right),\:\left(\mathrm{38},\:\mathrm{13},\:\mathrm{16}\right),\:\left(\mathrm{52},\:\mathrm{1},\:\mathrm{30}\right),\:\left(\mathrm{52},\:\mathrm{23},\:\mathrm{14}\right), \\ $$ $$\left(\mathrm{62},\:\mathrm{23},\:\mathrm{24}\right),\:\left(\mathrm{74},\:\mathrm{13},\:\mathrm{40}\right),\:\left(\mathrm{76},\:\mathrm{11},\:\mathrm{42}\right),\:\left(\mathrm{76},\:\mathrm{37},\:\mathrm{10}\right), \\ $$ $$\left(\mathrm{86},\:\mathrm{11},\:\mathrm{48}\right),\:\left(\mathrm{98},\:\mathrm{47},\:\mathrm{16}\right). \\ $$

Commented byAr 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 byBaliramKumar last updated on 13/Sep/22

thank you sir how taken time in programing see ↓⇊⇊

$${thank}\:{you}\:{sir}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$ $${how}\:{taken}\:{time}\:{in}\:{programing} \\ $$ $${see}\:\downarrow\downdownarrows\downdownarrows \\ $$

Commented byBaliramKumar last updated on 12/Sep/22

Commented byBaliramKumar last updated on 13/Sep/22

A tringle′s side in A.P. & area is rational (a, b, c) = (3, 4, 5), (13, 14, 15), (15, 26, 37) (17, 28, 39), (15, 28, 41), (25, 38, 51) (51, 52, 53), (29, 52, 75), (39, 62, 85) ..............

$${A}\:{tringle}'{s}\:\:{side}\:{in}\:{A}.{P}.\:\&\:{area}\:{is}\:{rational} \\ $$ $$\left({a},\:{b},\:{c}\right)\:=\:\left(\mathrm{3},\:\mathrm{4},\:\mathrm{5}\right),\:\:\left(\mathrm{13},\:\mathrm{14},\:\mathrm{15}\right),\:\:\left(\mathrm{15},\:\mathrm{26},\:\mathrm{37}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{17},\:\mathrm{28},\:\mathrm{39}\right),\:\left(\mathrm{15},\:\mathrm{28},\:\mathrm{41}\right),\:\left(\mathrm{25},\:\mathrm{38},\:\mathrm{51}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{51},\:\mathrm{52},\:\mathrm{53}\right),\:\left(\mathrm{29},\:\mathrm{52},\:\mathrm{75}\right),\:\left(\mathrm{39},\:\mathrm{62},\:\mathrm{85}\right) \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.............. \\ $$

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