Page 10: Equal Sums of Like Powers (Quartics)

New!:  Check out this author's "A Collection of Algebraic Identities" for more equal sums of like powers.

"Ramanujan and The Quartic Equation 2^4+2^4+3^4+4^4+4^4 = 5^4" The diophantine equation we can call a quartic sextuple, or a1^4 + a2^4 + a3^4 + a4^4 + a5^4 = a6^4, has only three known parametrizations, two of which were given by Ramanujan in terms of quadratic forms.  In this paper, we can show how to find an indefinite number of sextuple formulas by giving the relevant algebraic identity. Furthermore, this is also connected to Ramanujan's 6-10-8 Identity, and we give new versions of this, as well as for Hirschhorn's 3-7-5 Identity.  It seems the multiplicative properties of a^2+ab+b^2, an algebraic form connected to the Eisenstein integers, are responsible for the connection.

Choose your format:

RamQuad.htm

RamQuad.pdf

The continuation to fifth powers is in Page 14.

Side-article:

"There's Something About Sextuples..."

See also the new entries on:

http://mathworld.wolfram.com/Ramanujan6-10-8Identity.html

http://mathworld.wolfram.com/Hirschhorn3-7-5Identity.html

http://mathworld.wolfram.com/DiophantineEquation4thPowers.html

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This webpage was born Sept 22, 2005