# T. R. Parkin

**T. R. Parkin** (Thomas R. Parkin, per ProofWiki) was an American researcher who, with [L. J. Lander](https://www.edgepedia.org/l-j-lander), produced the first counterexample to [Euler's sum of powers conjecture](https://www.edgepedia.org/eulers-sum-of-powers-conjecture): the 1966 discovery that 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, found by a direct computer search on a CDC 6600.<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[2](https://proofwiki.org/wiki/Euler%27s_Sum_of_Powers_Conjecture/Historical_Note)</sup> This single result, published in a two-sentence note in the *Bulletin of the American Mathematical Society* and expanded in *Mathematics of Computation*, is what places Parkin in the mathematical record.<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup>

| Key fact | Detail |
|---|---|
| Signature result | Co-discoverer (with L. J. Lander) of 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, the smallest instance of four fifth powers summing to a fifth power and the first counterexample to Euler's conjecture<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup> |
| Publication | *Bull. AMS* 72(6), 1966, communicated by J. D. Swift on June 27, 1966; full paper in *Mathematics of Computation* received June 30, 1966<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> |
| Method | Direct search on the CDC 6600 using a precalculated table of fifth powers and the congruence x⁵ ≡ x (mod 30) to prune candidates<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> |
| Affiliations shown in print | Lander signed from Dartmouth College; Parkin has no affiliation line in the Bulletin note, and the *Math. Comp.* paper is signed from El Segundo, California<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> |
| Second paper | Co-author with Lander and J. L. Selfridge of "A Survey of Equal Sums of Like Powers" (*Math. Comp.* 21(99), 1967), which formulated the Lander–Parkin–Selfridge conjecture<sup>[4](https://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)</sup> |
| Biographical record | His education, dates, employer, and later career are undocumented; the El Segundo, California byline is the only workplace hint<sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> |

## Euler's conjecture and the 1966 counterexample

Euler conjectured, in a statement recorded in 1769, that at least n nth powers are required to sum to an nth power, for n > 2.<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup><sup> • </sup><sup>[5](https://oeis.org/A386541/internal)</sup> The conjecture generalizes the case n = 3 behind [Fermat's Last Theorem](https://www.edgepedia.org/fermats-last-theorem): it predicts that a fourth power cannot be a sum of fewer than four fourth powers, a fifth power not a sum of fewer than five fifth powers, and so on.

[The Bulletin](https://www.edgepedia.org/the-bulletin) note by Lander and Parkin, communicated by J. D. Swift on June 27, 1966, is famously short, two sentences: it reports that "a direct search on the CDC 6600 yielded 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ as the smallest instance in which four fifth powers sum to a fifth power," and that "this is a counterexample to a conjecture by Euler that at least n nth powers are required to sum to an nth power, n > 2."<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup> A full paper followed in *Mathematics of Computation*, received June 30, 1966 and revised July 29, 1966.<sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> MathWorld dates the disproof to 1967, presumably following the journal paper; the Bulletin note itself is from 1966.<sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup>

## How the counterexample was found

The search was computational, on the CDC 6600, and staged by case. The full paper describes the algorithm: a precalculated table of fifth powers replaced the taking of fifth roots in setting limits, and the congruence x⁵ ≡ x (mod 30) pruned candidates, since a fifth power modulo 30 is congruent to its own base.<sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup>

The stages, as reported in the paper, were:<sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup>

1. **n = 6, y ≤ 100**: ten primitive solutions found.
2. **n = 5, y ≤ 250**: four primitive solutions found; the fourth was "the unexpected result," 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, the counterexample.
3. **n = 4, y ≤ 750**: no further primitive solutions in that range.

The scale was modest by later standards. Generating all quadruples 1 ≤ a ≤ b ≤ c ≤ d ≤ 133 gives only about 13 million cases, checkable against a table of fifth powers, likely finishing in under an hour on the CDC 6600; the n = 4 search to 750⁵ covers over 5000 times as many cases.<sup>[7](https://mathoverflow.net/questions/325192/intuition-behind-counterexample-of-eulers-sum-of-powers-conjecture)</sup> Commenters on [MathOverflow](https://www.edgepedia.org/mathoverflow) observe that the authors' unelaborated "direct search" suggests no meet-in-the-middle method was needed, and that the Bulletin note's brevity likely reflects space limits, with the longer paper revealing the modular constraints used.<sup>[7](https://mathoverflow.net/questions/325192/intuition-behind-counterexample-of-eulers-sum-of-powers-conjecture)</sup>

ProofWiki adds an anecdote not documented elsewhere: Lander and Parkin were hunting for fifth powers that were sums of five fifth powers, and in one of the four solutions found, one contributing fifth power was 0⁵, at which point they realized they had a counterexample with only four terms.<sup>[2](https://proofwiki.org/wiki/Euler%27s_Sum_of_Powers_Conjecture/Historical_Note)</sup>

## The Lander–Parkin–Selfridge conjecture

In 1967 Lander, Parkin, and J. L. Selfridge published "A Survey of Equal Sums of Like Powers" in *Mathematics of Computation*, reporting a series of CDC 6600 searches to identify, for k < 10, the parameter sets (k, m, n) for which equal sums of like powers exist and to find the least solutions.<sup>[4](https://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)</sup> The survey formulates the conjecture now called the Lander–Parkin–Selfridge conjecture: for a positive-integer equality of sums of like powers with m terms on one side and n on the other, m + n ≥ k. MathWorld records that there are no known counterexamples to this conjecture, citing Ekl 1998.<sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup> So Parkin did work on it with Selfridge, as a co-author of the survey that stated it.

## Comparisons and later searches

**Elkies and the fourth-power case.** In 1986 Noam D. Elkies of Harvard University found the first counterexample for n = 4, using theoretical reasoning combined with a short computer search: 2,682,440⁴ + 15,365,639⁴ + 18,796,760⁴ = 20,615,673⁴.<sup>[8](https://www.sciencenews.org/article/eulers-sums-powers)</sup> MathWorld's table dates the 4.1.3 case to Elkies 1988, the year his paper appeared; Science News dates the discovery itself to 1986.<sup>[8](https://www.sciencenews.org/article/eulers-sums-powers)</sup><sup> • </sup><sup>[6](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)</sup> Fermat's Library's annotation notes that Elkies' solutions reduce to a parametric identity, giving an infinite family.<sup>[9](https://www.fermatslibrary.com/s/counterexample-to-eulers-conjecture-on-sums-of-like-powers)</sup> Roger Frye, at Thinking Machines, then used [Connection Machine](https://www.edgepedia.org/connection-machine) computers, working at night in his spare time, to find the smallest fourth-power solution: 95,800⁴ + 217,519⁴ + 414,560⁴ = 422,481⁴.<sup>[8](https://www.sciencenews.org/article/eulers-sums-powers)</sup>

**Fifth powers after 1966.** The 1967 survey records no further primitive (5.1.4) solutions up to 765⁵, and twelve primitive (5.1.5) solutions up to 599⁵, including Lander and Parkin's least solution 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵ = 72⁵; the y = 107 solution comes from Sastry's two-parameter identity.<sup>[4](https://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)</sup><sup> • </sup><sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup> A later computer search turned up a second fifth-power solution, 85359⁵ = 85282⁵ + 28969⁵ + 3183⁵ + 55⁵, and by 2000 seven fourth-power solutions were known for d < 2.1 × 10⁷.<sup>[8](https://www.sciencenews.org/article/eulers-sums-powers)</sup>

## By the numbers

The identity itself: 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, the smallest instance of four fifth powers summing to a fifth power.<sup>[1](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)</sup>

Search bounds and solution counts from the 1966–1967 papers:<sup>[3](https://doi.org/10.1090/s0025-5718-1967-0220669-3)</sup><sup> • </sup><sup>[4](https://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)</sup>

- 10 primitive n = 6 solutions with y ≤ 100.
- 4 primitive n = 5 solutions with y ≤ 250, the fourth being the counterexample.
- 12 primitive (5.1.5) solutions up to 599⁵.
- No further primitive (5.1.4) solutions up to 765⁵.

The contrast in computational scale is sharp. The 1966 search covered roughly 13 million quadruples, plausibly under an hour on a CDC 6600.<sup>[7](https://mathoverflow.net/questions/325192/intuition-behind-counterexample-of-eulers-sum-of-powers-conjecture)</sup> A preprint reporting the fourth primitive solution to a⁵ + b⁵ + c⁵ + d⁵ = e⁵ used a meet-in-the-middle strategy, enumerating and sorting sums of two fifth powers, at a cost of approximately 10,500,000 vCPU-hours over nine months.<sup>[10](https://arxiv.org/html/2603.05549v1)</sup>

## Recent developments

The main development in the subject area is the reported fourth primitive solution to a⁵ + b⁵ + c⁵ + d⁵ = e⁵, found by meet-in-the-middle search. Before this work only three primitive solutions were known: the 1966 Lander–Parkin solution, a 1996 solution with a negative term equal to 14132⁵, and a 2004 nonnegative solution equal to 85359⁵.<sup>[10](https://arxiv.org/html/2603.05549v1)</sup> The preprint's displayed equation reads 719115⁵ + 1331622⁵ + (−1340632)⁵ + 1956213⁵ = 1956878⁵.<sup>[10](https://arxiv.org/html/2603.05549v1)</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).](https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf)
2. [Euler's Sum of Powers Conjecture/Historical Note, ProofWiki.](https://proofwiki.org/wiki/Euler%27s_Sum_of_Powers_Conjecture/Historical_Note)
3. [L. J. Lander and T. R. Parkin (1967). A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation.](https://doi.org/10.1090/s0025-5718-1967-0220669-3)
4. [L. J. Lander, T. R. Parkin, J. L. Selfridge (1967). A Survey of Equal Sums of Like Powers. Mathematics of Computation 21(99).](https://www.ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/S0025-5718-1967-0222008-0.pdf)
5. [OEIS A386541.](https://oeis.org/A386541/internal)
6. [Euler's Sum of Powers Conjecture, Wolfram MathWorld.](https://mathworld.wolfram.com/EulersSumofPowersConjecture.html)
7. [MathOverflow: Intuition behind counterexample of Euler's sum of powers conjecture.](https://mathoverflow.net/questions/325192/intuition-behind-counterexample-of-eulers-sum-of-powers-conjecture)
8. [Euler's Sums of Powers, Science News.](https://www.sciencenews.org/article/eulers-sums-powers)
9. [Fermat's Library: annotated Counterexample to Euler's Conjecture.](https://www.fermatslibrary.com/s/counterexample-to-eulers-conjecture-on-sums-of-like-powers)
10. [The fourth known primitive solution to a^5+b^5+c^5+d^5=e^5 (arXiv preprint).](https://arxiv.org/html/2603.05549v1)

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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Number theorists › Computational number theorists*

*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*

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