Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of non-zero integers is itself a kth power, then n is greater than or equal to k.
Contents
In symbols, the conjecture falsely states that if
where n > 1 and a1, a2, …, an, b are non-zero integers, then n ≥ k.
The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case n = 2: if a1k + a2k = bk, then 2 ≥ k.
Although the conjecture holds for the case k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It is unknown whether the conjecture fails or holds for any value k ≥ 6.
Background
Euler had an equality for four fourth powers 594 + 1584 = 1334 + 1344; this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 33 + 43 + 53 = 63 or the taxicab number 1729. The general solution for:
is
where a and b are any integers.
Counterexamples
Euler's conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for k = 5. A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:
In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the k = 4 case. His smallest counterexample was
A particular case of Elkies' solutions can be reduced to the identity
where
This is an elliptic curve with a rational point at v1 = −31/467. From this initial rational point, one can compute an infinite collection of others. Substituting v1 into the identity and removing common factors gives the numerical example cited above.
In 1988, Roger Frye found the smallest possible counterexample
for k = 4 by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.
Generalizations
In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if k > 3 and
where ai ≠ bj are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m + n ≥ k. In the special case m = 1, the conjecture states that if
(under the conditions given above) then n ≥ k − 1.
The special case may be described as the problem of giving a partition of a perfect power into few like powers. For k = 4, 5, 7, 8 and n = k or k − 1, there are many known solutions. Some of these are listed below. There are no solutions for k = 6 where b ≤ 7005272580000000000♠272580.
k = 3
33 + 43 + 53 = 63 (Plato's number 216)
Srinivasa Ramanujan generalized it with x=1 and y=0 in formula