In 1769, Leonhard Euler conjectured a natural generalisation of Fermat's Last Theorem: that at least n nth powers are needed to sum to another nth power. For n=2, this is simply Fermat's observation about Pythagorean triples (3²+4²=5²), which requires exactly 2 squares. Euler's conjecture stood for nearly 200 years, apparently confirmed by every numerical check. Then in 1966, Lander and Parkin found a counterexample for n=5 — in a brief computer search producing a four-line paper, one of the shortest in mathematics history — showing 4 fifth powers can sum to a fifth power, contradicting Euler's conjecture directly.
Fermat's Last Theorem says you can't write a perfect cube as a sum of two cubes, or a fourth power as a sum of two fourth powers — you need at least three of the same power. Euler extended this idea: surely you need at least n nth powers to sum to an nth power? For squares, this is confirmed by Pythagorean triples (two squares summing to a square). For cubes and higher, it seemed to hold wherever anyone checked, for nearly two centuries.
Euler was perhaps the most prolific and accomplished mathematician in history, and he made this conjecture based on genuine mathematical reasoning — it's a natural generalisation, it agreed with every case anyone could check, and it had Fermat's theorem as a special case. And it was still completely wrong. Like the Mertens Conjecture (see that entry), it's a reminder that neither mathematical authority nor numerical evidence, however extensive, constitutes a proof.
Euler's conjecture (1769) states: if the sum of n kth powers equals a kth power, then n ≥ k. In other words, at least k terms are required on the left-hand side for the equation xₖ₁ + xₖ₂ + ... + xₖₙ = yᵏ to have a solution in positive integers. The case k=2 is trivially false (one square can equal another square: 4²=4²) if n=1, but the interesting content is that n=2 is possible (Pythagorean triples), which is the k=2 case of Fermat's observation. Fermat's Last Theorem is the statement that n=2 is impossible for k≥3 — a special case of Euler's conjecture, subsequently proven by Wiles.
Lander and Parkin's 1966 computer search on a CDC 6600 (one of the most powerful computers of the era) found the counterexample 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ — four fifth powers summing to a fifth power, contradicting the conjecture for k=5, n=4. Their paper announcing this was published in the Bulletin of the American Mathematical Society and is widely cited as one of the shortest mathematical papers ever published, consisting essentially of a statement of the problem, the counterexample, and a note on the search method.
For k=4 (fourth powers), the conjecture also turned out to be false — but this was discovered much later. Noam Elkies found in 1988, using a sophisticated theoretical approach (elliptic curve theory), that 2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴ — three fourth powers summing to a fourth power, violating the conjecture for k=4, n=3. Roger Frye subsequently found the smallest such counterexample for k=4 using a computer search guided by Elkies's theoretical insight.
The conjecture has been disproven for k=4 and k=5. For k=3 (cubes), it remains open whether 2 cubes can sum to a cube — but this is exactly Fermat's Last Theorem for cubes, proven by Euler himself (a special case of Wiles's 1995 proof). For k≥6, the conjecture remains neither proven nor disproven in general, though it is widely expected to be false based on the pattern of counterexamples found for lower k.
Noam Elkies's 1988 disproof for k=4 was not a blind computer search but a theoretically guided approach using the theory of elliptic curves (see the Clay Millennium Problems entry for the connection to the Birch and Swinnerton-Dyer Conjecture). Elkies showed that the existence of a solution to a²⁴ + b⁴ + c⁴ = d⁴ is equivalent to the existence of a rational point on a certain elliptic curve — and then found such a point using the arithmetic of elliptic curves. This approach, connecting a Diophantine problem to elliptic curve theory, is typical of modern number theory and illustrates how algebraic geometry provides tools for arithmetic problems that direct computation alone could not readily find.
The general problem of representing numbers as sums of kth powers — Waring's problem and its generalisations — is studied using the Hardy-Littlewood circle method (see the Goldbach Conjecture entry for a discussion in the additive prime context). The circle method provides asymptotic formulas for the number of representations of a large integer N as a sum of s kth powers, and it generally predicts that "generic" large integers are representable with few terms — suggesting that Euler's conjecture, which imposes a minimum number of terms, is unlikely to hold for large k, consistent with the pattern of disproofs found.
The 1966 Lander-Parkin result was one of the earliest significant mathematical discoveries made primarily by computer search, at a time when the role of computers in pure mathematics was still novel and somewhat contested. It is often cited alongside the 1976 computer-assisted proof of the Four Colour Theorem (see that entry) as an early landmark in the use of computers as tools for mathematical research — not merely calculation aids but genuine contributors to mathematical knowledge, though the philosophical questions raised by both results continue to be discussed (see the Proof entry for the broader context).