Fun with Pythagoras & the Pythagoreans

About: pythagorean natural numbers

Other than the Golden Verses (see this & that) there are Fermat like numbers for the 2nd degree such as 3,4,5  and n*(3,4,5)  what else is possible? Does it extend to a cube? Why not?
extending the pattern:
(3^3)+(4^3)+(5^3) = (6^3)  as well as (6^3)+(8^3)+(10^3)=(12^3) ; however apparently it doesn't extend to the 4th power.  See also diophantine equations at Wolfram. Is there another pattern beyond having the number of terms on the left be the same as the power which could help prove Fermat simply?

Tags

  1. pythagoras
  2. diophantine

See Also

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