
The Shortest Research Paper Ever Published - PLenz
https://sparkonit.com/2019/05/05/the-shortest-research-paper-ever-published
======
Someone
_”Euler proposed that the equation, a^n + b^n = c^n doesn’t hold true if the
value of “n” is greater than “2”. Then in 1966, two mathematicians L.J. Lander
and T.R. Parkin came along and swiftly overturned his claim with a
counterexample: 27^5 + 84^5 + 110^5 + 133^5 = 144^5”_

That’s mixing Euler’s claim that you need to sum at least n n-th powers to get
another n-th power
([https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...](https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture))
with the (less general) Fermat’s last theorem
([https://en.wikipedia.org/wiki/Fermat%27s_last_theorem](https://en.wikipedia.org/wiki/Fermat%27s_last_theorem))

The example in the paper disproved Euler’s conjecture, but did nothing about
Fermat’s.

------
andrepd
Euler proposed that the equation, a^n + b^n = c^n doesn’t hold true if the
value of “n” is greater than “2”. Then two mathematicians came along and
overturned his claim with a counterexample: 27^5 + 84^5 + 110^5 + 133^5 =
144^5

Is it me or there's something wrong with that paragraph? x)

------
astazangasta
>Euler proposed that the equation, an + bn = cn doesn’t hold true if the value
of “n” is greater than “2”.

This is wrong. That is Fermat's last theorem, which is true. Euler's
conjecture is related but more general:

>for all integers n and k greater than 1, if the sum of n kth powers of
positive integers is itself a kth power, then n is greater than or equal to k:

    
    
        a1k + a2k + ... + ank = bk ⇒ n ≥ k

------
hprotagonist
See also

    
    
      Conway and Soifer, “ Can n^2+ 1 unit equilateral triangles cover an equilateral triangle of side > n,say n + ε?” American Mathematical Monthly (2005).
    

which is two words and two figures.

Annotated so that it’s understandable: [https://fermatslibrary.com/s/shortest-
paper-ever-published-i...](https://fermatslibrary.com/s/shortest-paper-ever-
published-in-a-serious-math-journal-john-conway-alexander-soifer)

------
radioactivist
There is a story in a similar spirit about showing that one of the supposed
Mersenne primes (2^67-1) was in fact composite -- (probably apocryphally) by
simply doing the arithmetic by hand on a blackboard during a talk.

For more details, see:

\-
[https://en.wikipedia.org/wiki/Mersenne_prime](https://en.wikipedia.org/wiki/Mersenne_prime)

\- [https://hsm.stackexchange.com/questions/2105/whats-the-
famou...](https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-
about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106)

\- [https://mathoverflow.net/questions/207321/how-did-cole-
facto...](https://mathoverflow.net/questions/207321/how-did-cole-
factor-267-1-in-1903)

------
megalawn
Numberphile did a video on short papers:
[https://www.youtube.com/watch?v=QvvkJT8myeI](https://www.youtube.com/watch?v=QvvkJT8myeI)

------
elcomet
It's fun but not very interesting. A real research paper would explain the
method they used to find this examples. Even only a few lines if the method is
already known.

~~~
dahart
“A direct search on the CDC 6600 yielded”

That’s probably enough description of methodology for many people here to
reproduce the results, right? :)

~~~
mikeash
I wouldn’t even know where to start looking for a working CDC 6600.

~~~
dahart
I wouldn’t either. But I guess I don’t consider a CDC 6600 specifically to be
critical to the methodology. I just meant that a lot of people around here
could certainly write a computer program for direct search of the sum of four
fourth power integers. It seems like this might even be one of the problems on
Project Euler...?

~~~
mikeash
I was just pretending to misunderstand as a joke.

You did make me curious, though. The numbers are fairly small. I implemented
the dumbest possible brute force search for all combinations of numbers 1-255.
On my laptop, it found 133^5 + 110^5 + 84^5 + 27^5 = 144^5 in about 50
minutes.

I'd guess that this laptop is something like 100,000 times faster than the CDC
6600, so that dumb search would have taken about a decade. However, you can
get at least a couple orders of magnitude improvement with some simple
improvements.

~~~
pinewurst
I took it down to under 5 mins on my old Macbook by memoizing exponentiation
and aborting calculation/addition/comparison of later terms if the earlier sum
exceeded the result.

~~~
triska
Very nice! For comparison, here is a Prolog program that uses CLP(ℤ),
constraint logic programming over integers, to search for solutions:

    
    
        solution(Options, Vs) :-
                A^5 + B^5 + C^5 + D^5 #= E^5,
                Vs = [A,B,C,D,E],
                A #>= B, B #>= C, C #>= D,
                length(_, Exp),
                Upper #= 2^Exp,
                portray_clause(Upper),
                Vs ins 1..Upper,
                labeling(Options, Vs).
    

The speed of this program depends a lot on the Prolog implementation, and also
on the options that are used to specify the search strategy.

With this program, one can find a solution within one minute even with rather
slow Prolog systems on (current) commodity machines:

    
    
        ?- time(solution([bisect], Vs)).
        1.
        2.
        4.
        8.
        16.
        32.
        64.
        128.
        256.
        % 609,346,692 inferences, 57.855 CPU in 58.197 seconds (99% CPU, 10532282 Lips)
        Vs = [133, 110, 84, 27, 144] .

------
marttt
Here's an even shorter one:

Upper, Dennis. "The unsuccessful self-treatment of a case of “writer's
block”." Journal of Applied Behavior Analysis 7.3 (1974): 497.

[https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1311997/pdf/jab...](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1311997/pdf/jaba00061-0143a.pdf)

~~~
frosted-flakes
The reviewer comment is hilarious:

> _COMMENTS BY REVIEWER A_

> _I have studied this manuscript very carefully with lemon juice and X-rays
> and have not detected a single flaw in either design or writing style. I
> suggest it be published without revision. Clearly it is the most concise
> manuscript I have ever seen—yet it contains sufficient detail to allow other
> investigators to replicate Dr. Upper 's failure. In comparison with the
> other manuscripts I get from you containing all that complicated detail,
> this one was a pleasure to examine. Surely we can find a place for this
> paper in the Journal—perhaps on the edge of a blank page._

------
hervature
Link to a previous post that talks about short papers - including this one.
[https://news.ycombinator.com/item?id=15737611](https://news.ycombinator.com/item?id=15737611)

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selimthegrim
Surprised no one linked to this well-known preprint by Asher Peres.

[https://arxiv.org/abs/quant-ph/0310035](https://arxiv.org/abs/quant-
ph/0310035)

------
sys_64738
How do you discover these number in 1966?

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Upvoter33
sure, but the paper could have included the code ... :)

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jamesb93
That's just poor research.

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flywithdolp
It's probably about what we did to stop the global warming so far

