## x^2 + y^2 + z^2 = 3 xyz

Find 3 positive integers   $x, \; y$   and   $z$   which satisfy this equation:

Using this solution, find a different solution by changing one and only one of the numbers.

How many solutions does the equation have?

Extensions:

Are all triple solutions   $\{ \, x, \; y, \; z \, \}$    which contain a number,   $N$,    connected to each other on the tree found below?

Adding some triples to the tree:

$x = 1$

$\{1,1,2 \}$
$\{1,2,5 \}$
$\{1,5,13 \}$
$\{1,13,34 \}$
$\{1,34,89 \}$
$\{1,89,233 \}$
$\{1,233,610 \}$
$\{1,610,1597 \}$
$\{1,1597,4181 \}$
$\{1,4181,10946 \}$
$\{1,10946,28657 \}$
$\{1,28657,75025 \}$
$\{1,75025,196418 \}$
………………………………………..
…………………………………………

$x = 2$

$\{2,5,29\}$
$\{2,29,169\}$,   $\{2,169,985\}$
$\{2,985,5741\}$,   $\{2,5741,33461\}$
$\{2,33461,195025\}$,   $\{2,195025,1136689\}$
$\{2,1136689,6625109\}$,   $\{2,6625109,38613965\}$
$\{2,38613965,225058681\}$,   $\{2,225058681,1311738121\}$
$\{2,1311738121,7645370045\}$,   $\{2,7645370045,44560482149\}$
$\{2,44560482149,259717522849\}$,   $\{2,259717522849,1513744654945\}$
$\{2,1513744654945,8822750406821\}$,   $\{2,8822750406821,51422757785981\}$

………………………………………..
…………………………………………

$x = 5$

$\{5,29,433 \}$
$\{5,433,6466 \}$
$\{5,6466,96557 \}$

$\{5,96557,1441889 \}$
$\{5,1441889,21531778 \}$
$\{5,21531778,321534781 \}$

$\{5,321534781,4801489937 \}$
$\{5,4801489937,71700814274 \}$
$\{5,71700814274,1070710724173 \}$

$\{5,1070710724173,15988960048321 \}$
$\{5,15988960048321,238763690000642 \}$
$\{5,238763690000642,3565466389961309 \}$

$\{5,3565466389961309,53243232159418993 \}$
$\{5,53243232159418993,795083016001323586 \}$
$\{5,795083016001323586,11873002007860434797 \}$

$\{5,11873002007860434797,177299947101905198369 \}$
$\{5,177299947101905198369,2647626204520717540738 \}$
$\{5,2647626204520717540738,39537093120708857912701 \}$

$\{5,39537093120708857912701,590408770606112151149777 \}$
$\{5,590408770606112151149777,8816594465970973409333954 \}$
$\{5,8816594465970973409333954,131658508218958488988859533 \}$

$\{5,131658508218958488988859533,1966061028818406361423559041 \}$
$\{5,1966061028818406361423559041,29359256924057136932364526082 \}$
$\{5,29359256924057136932364526082,438422792832038647624044332189 \}$

$\{5,438422792832038647624044332189,6546982635556522577428300456753 \}$
$\{5,6546982635556522577428300456753,97766316740515800013800462519106 \}$
$\{5,97766316740515800013800462519106,1459947768472180477629578637329837 \}$

………………………………………..
…………………………………………

$x = 13$

$\{13,194,7561 \}$
$\{13,7561,294685 \}$
$\{13,294685,11485154 \}$
$\{13,11485154,447626321 \}$
$\{13,447626321,17445941365 \}$
$\{13,17445941365,679944086914 \}$
$\{13,679944086914,26500373448281 \}$
$\{13,26500373448281,1032834620396045 \}$
$\{13,1032834620396045,40254049821997474 \}$
$\{13,40254049821997474,1568875108437505441 \}$
$\{13,1568875108437505441,61145875179240714725 \}$
$\{13,61145875179240714725, 2383120256881950368834 \}$

………………………………………..
…………………………………………

$\{13,34,1325 \}$
$\{13,1325,51641 \}$
$\{13,51641,2012674 \}$
$\{13,2012674,78442645 \}$
$\{13,78442645,3057250481 \}$
$\{13,3057250481,119154326114 \}$
$\{13,119154326114,4643961467965 \}$
$\{13,4643961467965,180995342924521 \}$
$\{13,180995342924521,7054174412588354 \}$
$\{13,7054174412588354,274931806748021285 \}$
$\{13,274931806748021285,10715286288760241761 \}$
$\{13,10715286288760241761,417621233454901407394 \}$

………………………………………..
…………………………………………

$x = 29$

$\{29,169,14701 \}$
$\{29,14701,1278818 \}$
$\{29,1278818,111242465 \}$
$\{29,111242465,9676815637 \}$
$\{29,9676815637,841771717954 \}$
$\{29,841771717954,73224462646361 \}$
$\{29,73224462646361,6369686478515453 \}$
$\{29,6369686478515453,554089499168198050 \}$
$\{29,554089499168198050,48199416741154714897 \}$

………………………………………..
…………………………………………

$\{29,433,37666 \}$
$\{29,37666,3276509 \}$
$\{29,3276509,285018617 \}$
$\{29,285018617,24793343170 \}$
$\{29,24793343170,2156735837173 \}$
$\{29,2156735837173,187611224490881 \}$
$\{29,187611224490881,16320019794869474 \}$
$\{29,16320019794869474,1419654110929153357 \}$
$\{29,1419654110929153357,123493587631041472585 \}$

………………………………………..
…………………………………………

the next values for   $x$   are   $34, \; 89$