This paper is one of the shortest papers ever published by a seriou...
In 1966 Lander and Parkin through a direct computer search on a CDC...
In 1986, Noam Elkies found a method to construct an infinite series...
i
9
66] COUNTEREXAMPLE TO EULER'S CONJECTURE 1079
2.
F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2)
30 (1930), 264-286.
DARTMOUTH COLLEGE
COUNTEREXAMPLE TO EULER'S CONJECTURE
ON SUMS OF LIKE POWERS
BY L. J. LANDER AND T. R. PARKIN
Communicated by J. D. Swift, June 27, 1966
A direct search on the CDC 6600 yielded
27
5
+ 84
5
+ HO
5
+ 133
6
- 144
5
as the smallest instance in which four fifth powers sum to a fifth
power. This is a counterexample to a conjecture by Euler [l] that at
least n nth powers are required to sum to an nth power, n>2.
REFERENCE
1.
L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York,
1952,
p. 648.

Discussion

This paper is one of the shortest papers ever published by a serious Mathematical Journal. L.J. Lander and T.R. Parkin show a counterexample of Euler's sum of powers conjecture. Euler conjectured in 1769 that at least $n$ $n^{th}$ powers are required for $n>2$ to provide a sum that is itself an $n^{th}$ power. In other words if $\sum_{i=1}^{n}a_i^k = b^k$ where $n>2$ and $a_1, a_2, \dots, a_n, b$ are non-zero integers, then $n\geq k$. The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case $n = 2$: if $a_1^k + a_2^k = b^k$, then $2 \geq k$. In 1966 Lander and Parkin through a direct computer search on a CDC 6600, found a counterexample for k = 5. ![CDC 6600](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/CDC_6600.jc.jpg/1920px-CDC_6600.jc.jpg) There's a detailed discussion [here](http://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf) of how the authors approached the problem from a computational standpoint. In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the $k = 4$ case. His smallest counterexample was $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$. A particular case of Elkies' solutions can be reduced to the identity. $(85v^2 + 484v − 313)^4 + (68v^2 − 586v + 10)^4 + (2u)^4 = (357v^2 − 204v + 363)^4$ where $u^2 = 22030 + 28849v − 56158v^2 + 36941v^3 − 31790v^4$. 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.