Euler's Conjecture

"It has seemed to many Geometers that this theorem[Fermat's Last Theorem] may be generalized. Just as there do not exist two cubes whose sum or difference is a cube, it is certain that it is impossible to exhibit three biquadrates whose sum is a biquadrate, but that at least four biquadrates are needed if their sum is to be a biquadrate, although no one has been able up to the present to assign four such biquadrates. In the same manner it would seem to be impossible to exhibit four fifth powers whose sum is a fifth power, and similarly for higher powers."

No matter how great a mathematician Euler was, there was no guarantee that the conjecture was true. In fact, the reverse happened. Truth was slowly ripped off his conjecture as mathematicians began to disprove it.

No advancement was made on Euler's statement until 1911 when R. Norrie assigned four such biquadrates:

Fifty-five years later a computer search by Parking and Lander gave a counter-example to Euler's more general conjecture:

Then in the 1988 issue of Mathematics of Computation, Noam Elkies published a paper to disprove Euler's conjecture for fourth powers. He found a way to generate infinite family of solutions, of which the first member is

On A⁴ + B⁴ + C⁴ = D⁴

Since that Diophantine Equation is homogenous, solving it is equivalent to finding a point r, s, t = ( A/D, B/D, C/D) on the surface of r⁴ + s⁴ + t⁴ = 1 with rational coordinates r, s, t. We first start with an analysis of a parametrization of r⁴ + s⁴ + t⁴ = 1 as a pencil of conics. This yields a parametrization of r⁴ + s⁴ + t⁴ = 1 as a pencil of curves of genus one.

Then we find the simplest curve in the pencil which could possibly have a rational point that would disprove Euler's conjecture. It happens that there is such a rational point of sufficiently small height to have been found by a direct computer search : first solution -

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴

Reference: Noam Elkies, On A⁴ + B⁴ + C⁴ = D⁴, Mathematics of Computation, v51(1988) P.825-835

The solution comes from an elliptic curve given by u = -5 / 8 in the parametric solution

of x⁴ - y⁴ = z⁴ + t⁴ given by Dem'janenko. As there are infinitely many values of u which give curves of positive rank, there are thus further families of solutions. This is, however, not the smallest solution. In Elkies' article, he said :

While our first counter-example

(A, B, C; D) = (2682440, 15365639, 18796760; 20615673)

to Euler's conjecture still seems beyond the range of reasonable exhaustive computer search, there remained the possibility that smaller solutions may be found by such a search. Shortly after hearing of the first solution, Roger Frye of Thinking Machines Corporation asked whether it was minimal; I did not know, but suggested how one might exhaustively search for smaller solutions : eliminating common factors and permuting A, B, C if necessary, we may take D odd and not divisible by 5, and C < D such that D⁴ - C⁴ is divisible by 625 and satisfies several other congruence and divisibility properties, and for each such D and C look for a representation of D⁴ - C⁴ as A⁴ + B⁴ with A, B divisible by 5. Frye translated this into a computer program and ran it on various Connection Machines for about 100 hours to find the minimal counter-example to Euler's conjecture:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴

Despite the success in finding counter-examples for 4^th and 5^th powers, solutions of higher powers are still unknown. Was Euler totally wrong, or is there still some truth lying in his conjecture?