Editing Pythagorean triple (section)

==={{math|''n'' − 1}} or {{math|''n''}} {{math|''n''}}th powers summing to an {{math|''n''}}th power===
{{main|Euler's sum of powers conjecture}}
Another generalization is searching for sequences of {{math|1=''n'' + 1}} positive integers for which the {{math|''n''}}th power of the last is the sum of the {{math|''n''}}th powers of the previous terms. The smallest sequences for known values of {{math|''n''}} are:

* {{math|''n''}} = 3: {3, 4, 5; 6}.
* {{math|''n''}} = 4: {30, 120, 272, 315; 353}
* {{math|''n''}} = 5: {19, 43, 46, 47, 67; 72}
* {{math|''n''}} = 7: {127, 258, 266, 413, 430, 439, 525; 568}
* {{math|''n''}} = 8: {90, 223, 478, 524, 748, 1088, 1190, 1324; 1409}

For the {{math|1=''n'' = 3}} case, in which <math>x^3+y^3+z^3=w^3,</math> called the  [[Fermat cubic]], a general formula exists giving all solutions.

A slightly different generalization allows the sum of {{math|(''k'' + 1)}} {{math|''n''}}th powers to equal the sum of {{math|(''n'' − ''k'')}} {{math|''n''}}th powers. For example:
* ({{math|1=''n'' = 3}}): 1{{sup|3}} + 12{{sup|3}} = 9{{sup|3}} + 10{{sup|3}}, made famous by Hardy's recollection of a conversation with [[Ramanujan]] about the number [[Taxicab number|1729]] being the smallest number that can be expressed as a sum of two cubes in two distinct ways.

There can also exist {{math|''n'' − 1}} positive integers whose {{math|''n''}}th powers sum to an {{math|''n''}}th power (though, by [[Fermat's Last Theorem]], not for {{math|1=''n'' = 3)}}; these are counterexamples to [[Euler's sum of powers conjecture]].  The smallest known counterexamples are<ref>{{citation |first1=Scott |last1=Kim |author-link1=Scott Kim|title=Bogglers |journal=[[Discover (magazine)|Discover]] |page=82 |date=May 2002 |url=http://discovermagazine.com/2002/may/bogglers |quote=The equation w{{sup|4}} + x{{sup|4}} + y{{sup|4}} = z{{sup|4}} is harder. In 1988, after 200 years of mathematicians' attempts to prove it impossible, [[Noam Elkies]] of Harvard found the counterexample, 2,682,440{{sup|4}} + 15,365,639{{sup|4}} + 18,796,760{{sup|4}} = 20,615,673{{sup|4}}.}}</ref><ref>{{citation |author-link=Noam Elkies |last=Elkies |first=Noam |title=On A{{sup|4}} + B{{sup|4}} + C{{sup|4}} = D{{sup|4}} |journal=Mathematics of Computation |volume=51 |issue=184 |pages=825–835 |year=1988 |mr=930224 |url=https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-1988-0930224-9/ | doi = 10.2307/2008781|jstor=2008781 }}</ref><ref name=MacHale>{{citation |last1=MacHale |first1=Des |author1-link=Des MacHale|last2=van den Bosch |first2=Christian |title=Generalising a result about Pythagorean triples |journal=[[Mathematical Gazette]] |volume=96 |pages=91–96 |date=March 2012 |doi=10.1017/S0025557200004010 |s2cid=124096076 |doi-access=free }}</ref>

* {{math|1=''n'' = 4}}: (95800, 217519, 414560; 422481)
* {{math|1=''n'' = 5}}: (27, 84, 110, 133; 144)