# Euler's sum of powers conjecture

**Euler's sum of powers conjecture** is a disproved conjecture of [Leonhard Euler](https://www.edgepedia.org/leonhard-euler), stated in 1769, that at least \( n \) nth powers are required to sum to an nth power for \( n > 2 \). It generalizes [Fermat's Last Theorem](https://www.edgepedia.org/fermats-last-theorem), which is the case of two terms. A direct computer search on a CDC 6600 by [L. J. Lander](https://www.edgepedia.org/l-j-lander) and T. R. Parkin refuted it in 1966 with \( 27^5 + 84^5 + 110^5 + 133^5 = 144^5 \), and Noam Elkies disproved the fourth-power case in 1988.<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup><sup> • </sup><sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup>

| Key fact | Detail |
|---|---|
| Statement | For \( n > 2 \), at least \( n \) nth powers are required to sum to an nth power; equivalently, no nth power is a sum of fewer than \( n \) smaller nth powers<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup> |
| Origin | Conjectured by Euler in 1769 as a generalization of Fermat's Last Theorem<sup>[3](https://www.cs.man.ac.uk/~rizos/EqualSums/equalsums.html)</sup> |
| First counterexample | \( 27^5 + 84^5 + 110^5 + 133^5 = 144^5 \), found in 1966 by Lander and Parkin on a CDC 6600<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup> |
| Publication | One page, *Bulletin of the American Mathematical Society* 72(6), p. 1079; communicated by J. D. Swift, June 27, 1966<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup> |
| Fourth-power case | Disproved by Noam Elkies in 1988, who showed \( A^4 + B^4 + C^4 = D^4 \) has infinitely many positive integer solutions<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup> |
| Smallest quartic counterexample | \( 95800^4 + 217519^4 + 414560^4 = 422481^4 \), found by Frye and proven minimal by exhaustive search<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup> |
| Open status | No counterexamples are known for exponents above 5<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup> |

## The conjecture

In 1769 Euler, generalizing Fermat's Last Theorem, conjectured that "it is impossible to exhibit three fourth powers whose sum is a fourth power", "four fifth powers whose sum is a fifth power, and similarly for higher powers".<sup>[3](https://www.cs.man.ac.uk/~rizos/EqualSums/equalsums.html)</sup> In the form recorded in the 1966 paper, the claim is that at least \( n \) nth powers are required to sum to an nth power for \( n > 2 \).<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup> The cubic case \( n = 3 \) coincides with the case \( n = 3 \) of Fermat's Last Theorem, so the conjecture's first genuinely new case was \( n = 4 \): that no fourth power is the sum of three fourth powers.<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup><sup> • </sup><sup>[4](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)</sup>

The conjecture sits in the same family of additive questions as [Waring's problem](https://www.edgepedia.org/warings-problem), which asks how many kth powers are needed to represent every integer; Euler's question instead fixes the target to be a single kth power and asks how few terms can suffice.<sup>[5](https://mathworld.wolfram.com/WaringsProblem.html)</sup>

## The 1966 counterexample

Lander and Parkin, then at the Aerospace Corporation, ran a direct search on a Control Data Corporation CDC 6600 and found

\[ 27^5 + 84^5 + 110^5 + 133^5 = 144^5 \]

as the smallest instance in which four fifth powers sum to a fifth power.<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup> The announcement, "Counterexample to Euler's Conjecture on Sums of Like Powers", was communicated by J. D. Swift on June 27, 1966 and occupies a single page of the *Bulletin of the American Mathematical Society* (volume 72, issue 6, page 1079); its only reference is to Dickson's *History of the Theory of Numbers*.<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup>

Dating varies slightly across the literature: the bulletin page itself is dated 1966, while MathWorld and the journal *Mathematics of Computation* credit the disproof to Lander and Parkin (1967), the year of their full paper "A Counterexample to Euler's Sum of Powers Conjecture" in *Mathematics of Computation* 21, pp. 101–103.<sup>[1](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)</sup><sup> • </sup><sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup> The counterexample is listed in the OEIS as sequence A386541, the integers 27, 84, 110, 133, 144.<sup>[7](https://oeis.org/A386541)</sup>

The search arose from a broader computational program. In 1967 Lander, Parkin, and Selfridge published "A Survey of Equal Sums of Like Powers" in *Mathematics of Computation*, attacking the general [Diophantine equation](https://www.edgepedia.org/diophantine-equation) \( x_1^k + \cdots + x_m^k = y_1^k + \cdots + y_n^k \) and seeking, for each \( k \) and \( m \), the least \( n \) for which it is solvable.<sup>[3](https://www.cs.man.ac.uk/~rizos/EqualSums/equalsums.html)</sup><sup> • </sup><sup>[8](https://www.ams.org/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)</sup>

## The fourth-power case

The conjecture survived in its original motivating form, the "quartic conjecture" that \( A^4 + B^4 + C^4 = D^4 \) has no positive integer solutions, until 1988. [Noam Elkies](https://www.edgepedia.org/noam-elkies), a Harvard-trained mathematician who received his PhD at age twenty, transformed the problem into one about rational points on elliptic curves: integer solutions of the quartic equation correspond to rational solutions of \( r^4 + s^4 + t^4 = 1 \), and Elkies parametrized a pencil of conics to produce elliptic curves whose rational points could be searched by computer.<sup>[4](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)</sup> A rational point of infinite order on one such curve gave the counterexample

\[ 2682440^4 + 15365639^4 + 18796760^4 = 20615673^4 \]

and, because the point has infinite order, infinitely many counterexamples.<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup><sup> • </sup><sup>[4](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)</sup> Elkies also showed the rational solutions are dense in the real solutions of the normalized equation.<sup>[4](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)</sup>

Shortly afterward, Frye found the smallest counterexample, \( 95800^4 + 217519^4 + 414560^4 = 422481^4 \), which was proven minimal by exhaustive computer search.<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup><sup> • </sup><sup>[4](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)</sup> The gap in size between Elkies' first solution and the minimal one illustrates why the conjecture resisted two centuries of hand computation: the smallest quartic counterexample has a right-hand side above \( 4 \times 10^5 \), and further solutions grow quickly, with a 2007 brute-force search for \( d < 2 \times 10^9 \) finding the first 19 solutions and more than 100 primitive solutions now known below \( d < 10^{27} \).<sup>[9](https://math.stackexchange.com/questions/5132298/recent-counter-examples-to-eulers-sum-of-powers-conjecture?noredirect=1)</sup>

## Fifth powers and the extended conjecture

For fifth powers, the Lander–Parkin identity remained for decades nearly alone. In 2004 Frye obtained the only other known primitive solution, \( 55^5 + 3183^5 + 28969^5 + 85282^5 = 85359^5 \).<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup> Further fifth-power counterexamples have continued to appear; a 2026 example by Jeffrey Braun uses a negative term, \( (-1340632)^5 + 719115^5 + 1331622^5 + 1956213^5 = (11 \times 177898)^5 \).<sup>[9](https://math.stackexchange.com/questions/5132298/recent-counter-examples-to-eulers-sum-of-powers-conjecture?noredirect=1)</sup>

Randy Ekl reformulated the surviving question in 1998 as an extended conjecture about the equation \( a_1^k + \cdots + a_m^k = b_1^k + \cdots + b_n^k \): that no solutions exist with \( m + n < k \). Defining \( \Delta_k = \min_{m,n}(m + n - k) \) over all known solutions, the conjecture asserts \( \Delta_k \ge 0 \), and no counterexamples to it are known.<sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup> Successive computational work has pushed known \( \Delta_k \) values toward zero, for example S. Chase's 8.3.5 solution with \( \Delta_8 = 0 \) and Ekl's 9.4.6 and 9.5.5 solutions with \( \Delta_9 = 1 \).<sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup>

## Current status

The original conjecture is false for exponents 4 and 5. For exponents above 5, no solutions to the relevant equations are known, and whether counterexamples exist remains open.<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup><sup> • </sup><sup>[10](https://mathdb.com/p/346540/euler-s-sum-of-powers-conjecture)</sup> Variants of the question continue to be studied; a 2021 paper by Tianxin Cai and Yong Zhang used the theory of elliptic curves to analyze a Diophantine system whose solutions include those of Euler's equation, proving among other results that the system has no positive integer solution for \( s = 3 \) but infinitely many for \( s > 4 \).<sup>[2](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)</sup>

## References

1. [L. J. Lander and T. R. Parkin (1966). Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society 72(6), 1079.](https://marktomforde.com/academic/miscellaneous/images/ShortestPaper.pdf)
2. [Tianxin Cai and Yong Zhang (2021). A Variety of Euler's Sum of Powers Conjecture. Czechoslovak Mathematical Journal 71(146), 1099–1113.](https://dml.cz/bitstream/handle/10338.dmlcz/149240/CzechMathJ_71-2021-4_13.pdf)
3. [Rizos Sakellariou (1998). On Equal Sums of Like Powers (Euler's Conjecture), University of Manchester.](https://www.cs.man.ac.uk/~rizos/EqualSums/equalsums.html)
4. [Teaching Elliptic Curves Using Original Sources (Elkies' quartic counterexample), MAA, From Calculus to Computers.](https://maa.org/wp-content/uploads/2024/10/NTE68.pdf)
5. [Waring's Problem, Wolfram MathWorld.](https://mathworld.wolfram.com/WaringsProblem.html)
6. [Euler's Sum of Powers Conjecture, Wolfram MathWorld.](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)
7. [OEIS A386541: Lander and Parkin's 1966 counterexample to Euler's sum of powers conjecture.](https://oeis.org/A386541)
8. [L. J. Lander, T. R. Parkin, and J. L. Selfridge (1967). A Survey of Equal Sums of Like Powers. Mathematics of Computation 21, 446–459.](https://www.ams.org/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)
9. [Recent counter-examples to Euler's Sum of Powers conjecture? Mathematics Stack Exchange.](https://math.stackexchange.com/questions/5132298/recent-counter-examples-to-eulers-sum-of-powers-conjecture?noredirect=1)
10. [Euler's sum of powers conjecture, MathDB.](https://mathdb.com/p/346540/euler-s-sum-of-powers-conjecture)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Additive number theory › Waring's problem and Hilbert–Waring theory*

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