- 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 thePythagorean s before him, that 6 was the firstperfect number . Furthermore, he was also aware that the sum of the cubes of the sides of the 3-4-5triangle 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 byRamanujan ,:egin{align}& (3x^2+5xy-5y^2)^3+(4x^2-4xy+6y^2)^3+(5x^2-5xy-3y^2)^3 \& = (6x^2-4xy+4y^2)^3.end{align}
This turns out to be just a special case of the more general,
:egin{align}& (ax^2-v_1xy+by^2)^3 + (bx^2+v_1xy+ay^2)^3 \& {} + (cx^2+v_2xy+dy^2)^3 + (dx^2-v_2xy+cy^2)^3 \& = (a^3+b^3+c^3+d^3)(x^2+v_3y^2)^3,end{align}
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 number Fact|date=July 2008References
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]
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