Euler(5.2.2)
It is conjectured that a5+b5=c5+d5 equation has no non-trivial integer solutions - see also Taxicab(5,2,2). In 2005 T. Piezas found the following parameterization:
"(√p+√q)5 + (√p-√q)5 = (√r+√s)5 + (√r-√s)5,
{p,q,r,s} = {5vw2, -1+uw2, 5v, -(u+10v)+w3}, where w = u2+10uv+5v2
One can then set v = 5t2 so that {p,r} are squares. Whether {q,s} can then be made non-trivial squares is another matter."
I used these formulas to create simple program in C++ to search for integer solutions of the initial equation.
Observations used to speed the program up:
q has to be square and q=-1+uw2, so for a given u one can just solve special case of Pell equation: g2-uw2=-1 to get w, and finally to get v
squares can have only selected remainders modulo given natural number. That can be applied to formulas for q and s to get allowed set of u modulo given number. In practice, about 98,86% of u values can be skipped, if using arithmetic modulo 100800.
Results:
A) No integer solutions found. Tested areas:
u ≤ 3*1011 and w ≤ 23000
u ≤ 1013 and w ≤ 2300
B) Partial result:
in tested areas q is square only for {u,v} = {1+k2, k2}, where k is any integer number, so obviously v is not of the 5t2 form.
C) Assuming above parameterization is general a lower bound for the Taxicab problem is:
Taxicab(5,2,2) > 10162