Plato's number

Plato's number is the number 216 = 6³ alluded to in an obscure passage in "The Republic", Book VIII. [cite journal|last=Donaldson|first=Rev. J. W|title=On Plato's Number|journal=Proceedings of the Philological Society|volume=1|issue=8|pages=81–90|date=April 7, 1843|publisher=Philological Society (Great Britain)] In this passage, Plato discusses the properties of a certain number and remarks that this is "...on which would depend the better and worse generations in his imaginary republic" (Donaldson). He knew, like the Pythagoreans before him, that 6 was the first perfect number. Furthermore, he was also aware that the sum of the cubes of the sides of the 3-4-5 triangle was equal to

:$3^3+4^3+5^3=6^3.,$

Related identities

Plato's number expressed as 3³ + 4³ + 5³ = 6³ leads to the identity found by Ramanujan,

:

This turns out to be just a special case of the more general,

:

where

:$v\_1\; =\; c^2-d^2,\; v\_2\; =\; a^2-b^2,\; ext\{\; and\; \}v\_3\; =\; (a+b)(c+d).$

Thus it remains to solve "a"³ + "b"³ + "c"³ + "d"³ ="K", where "K" is zero or any number of cubes.

Nice results involving cubes in arithmetic progression are given by

:$11^3+12^3+13^3+14^3\; =\; 20^3,,$

:$31^3+33^3+35^3+37^3+39^3+41^3\; =\; 66^3,,$

and others found by solving a certain elliptic curve.

ee also

* Norrie's numberFact|date=July 2008

References

External links

*
* [http://www.geocities.com/titus_piezas/ramanujan_page9.html Ramanujan And The Cubic Equation 3³ + 4³ + 5³ = 6³]
* [http://www.math.niu.edu/~rusin/known-math/97/cube.sum Sum of Consecutive Cubes Equals a Cube]