User blog:AJZajac/Tetragorean Tuples

​I've played around with a tetrational version of Pythagorean triples. These "tetragorean tuples"  decompose a number raised to its own power (let's call this a two-tower) into the product of four or more m^m = a^a b^b c^c ...

The smallest two-tower that can form a tetragorean tuple is 24^24, which can be represented as the product of four two-towers:

24^24 = (2^2)(6^6)(9^9)(16^16)

Here are some others involving two-towers with base < 100:

36^36 = (6^6)(8^8)(9^9)(12^12)(18^18)

60^60 = (3^3)(4^4)(5^5)(6^6)(15^15)(18^18)(20^20)(30^30)

72^72 = (3^3)(12^12)(27^27)(48^48)

78^78 = (2^2)(3^3)(4^4)(8^8)(13^13)(18^18)(26^26)(39^39)

84^84 = (2^2)(8^8)(9^9)(14^14)(24^24)(28^28)

96^96 = (3^3)(4^4)(12^12)(16^16)(27^27)(64^64)

As with Pythagorean triples, there are values that can be decomposed in more than one way. Also, I think it's safe to conjecture that there are no tuples that can be formed based upon three-towers or higher levels of tetration (i.e. m^^3 = a^^3 b^^3 c^^3...). This shouldn't be as tough to prove as Fermat's Last Theorem. ;)