<i>”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”</i><p>That’s mixing Euler’s claim that you need to sum at least n n-th powers to get another n-th power (<a href="https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture" rel="nofollow">https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjec...</a>) with the (less general) Fermat’s last theorem (<a href="https://en.wikipedia.org/wiki/Fermat%27s_last_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Fermat%27s_last_theorem</a>)<p>The example in the paper disproved Euler’s conjecture, but did nothing about Fermat’s.
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<p>Is it me or there's something wrong with that paragraph? x)
>Euler proposed that the equation, an + bn = cn doesn’t hold true if the value of “n” is greater than “2”.<p>This is wrong. That is Fermat's last theorem, which is true. Euler's conjecture is related but more general:<p>>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:<p><pre><code> a1k + a2k + ... + ank = bk ⇒ n ≥ k</code></pre>
See also<p><pre><code> Conway and Soifer, “ Can n^2+ 1 unit equilateral triangles cover an equilateral triangle of side > n,say n + ε?” American Mathematical Monthly (2005).
</code></pre>
which is two words and two figures.<p>Annotated so that it’s understandable: <a href="https://fermatslibrary.com/s/shortest-paper-ever-published-in-a-serious-math-journal-john-conway-alexander-soifer" rel="nofollow">https://fermatslibrary.com/s/shortest-paper-ever-published-i...</a>
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.<p>For more details, see:<p>- <a href="https://en.wikipedia.org/wiki/Mersenne_prime" rel="nofollow">https://en.wikipedia.org/wiki/Mersenne_prime</a><p>- <a href="https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106" rel="nofollow">https://hsm.stackexchange.com/questions/2105/whats-the-famou...</a><p>- <a href="https://mathoverflow.net/questions/207321/how-did-cole-factor-267-1-in-1903" rel="nofollow">https://mathoverflow.net/questions/207321/how-did-cole-facto...</a>
Numberphile did a video on short papers: <a href="https://www.youtube.com/watch?v=QvvkJT8myeI" rel="nofollow">https://www.youtube.com/watch?v=QvvkJT8myeI</a>
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.
Here's an even shorter one:<p>Upper, Dennis. "The unsuccessful self-treatment of a case of “writer's block”." Journal of Applied Behavior Analysis 7.3 (1974): 497.<p><a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1311997/pdf/jaba00061-0143a.pdf" rel="nofollow">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1311997/pdf/jab...</a>
Link to a previous post that talks about short papers - including this one. <a href="https://news.ycombinator.com/item?id=15737611" rel="nofollow">https://news.ycombinator.com/item?id=15737611</a>
Surprised no one linked to this well-known preprint by Asher Peres.<p><a href="https://arxiv.org/abs/quant-ph/0310035" rel="nofollow">https://arxiv.org/abs/quant-ph/0310035</a>