Page 13: The Fermat Surface a^k + b^k = c^k + d^k

"Taxicab Numbers and the Fermat Surface a^k+b^k = c^k+d^k for k = 4,6,8" The equation a^k+b^k = c^k+d^k for k =3 has a famous anecdote connected to it, namely the incident involving Ramanujan, Hardy, and the taxicab number 1729 (as if you didn't know already).

An example of a turn-of-the-century taxicab which might have been similar to the one ridden by Hardy. Note the number "5795".  For more images, visit the London Vintage Taxi Association.

In a previous paper we extended Euler's method of finding the complete rational parametrization of k = 3 to find a radical version for k = 5.  For this one, the generalization will be completed so that it can be applied for any k.  We will then:     a) give a solution to the above equation for k = 4 using fifth powers.     b) find a radical four-parameter solution to k = 6 which for certain parameters will be multi-grade for k = 2,4,6.     c) a radical solution to k = 8 that may involve, surprisingly, the Golden Ratio and the Plastic Constant.

Choose your format: (preferably the pdf)

Fermatsurface.pdf

Fermatsurface.htm

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This webpage was born Nov 29, 2005