"The equation a^k+b^k+c^k = d^k has no non-trivial integral solution for k = 5. I have discovered a truly marvelous demonstration of this proposition that this website is too narrow to contain..."
Seriously, this paper will give a four-parameter solution in the radicals to the above equation. Only one other formula for this is known, a complex one found by Desboves in 1880 and re-discovered by Noam Elkies in 1995. The new one may be the complete parametrization of such quintic quadruples as it was derived using the same method Euler used to find the one for a^3+b^3+c^3 = d^3 (with a twist).
In general, this is about Euler's Extended Conjecture, a generalization of FLT. We will give the background information about this plus make a summary of known numerical results. A timeline is also available in Page 12.
