x^2 + y^2 + z^2 = 3 xyz

 
 

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

                         equa

 

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

How many solutions does the equation have?  

Source:   
http://galileo.org/classroom-examples/math/math-fair-problems/puzzles/xyz-tree/

 

Extensions:

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

xyz tree

 
 
 

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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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