The Shortest Research Paper Ever Published
sparkonit.com
sparkonit.com
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...) with the (less general) Fermat’s 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.
Is it me or there's something wrong with that paragraph? x)
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 Conway and Soifer, “ Can n^2+ 1 unit equilateral triangles cover an equilateral triangle of side > n,say n + ε?” American Mathematical Monthly (2005).
Annotated so that it’s understandable: https://fermatslibrary.com/s/shortest-paper-ever-published-i...
For more details, see:
- https://en.wikipedia.org/wiki/Mersenne_prime
- https://hsm.stackexchange.com/questions/2105/whats-the-famou...
- https://mathoverflow.net/questions/207321/how-did-cole-facto...
That’s probably enough description of methodology for many people here to reproduce the results, right? :)
There's a C one. Truly though, it wouldn't be hard to just write a brute force checker for this in some reasonably efficient compiled language.
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.
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).
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] .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...
> 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.