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