Shortest known paper published

shortest-math-paper.jpg

I recently saw a post from OpenCulture, that I explored, and tweeted about: the shortest known paper published in a serious math journal.

I wrote few lines of C++ code to search for four numbers less than 144 whose fifth powers sum up to 144^5.

#include <iostream>
#include <cmath>
using namespace std; 

// Disprove Euler conjecture
// Babis Tsourakakis (babis@seas.harvard.edu)
// March 22, 2017

long min(long x,long y){
    if(x<=y) return x;
    else return y;
}

void disprove_euler(long N){
	long target = pow(N,5);
        for(long x = N-1;x>= floor( 0.25*N ); x--){
     for(long y = floor( pow( target - pow(x,5), 0.2)); y>=1;y--){
			for(long z = min(y, floor( pow(target - pow(x,5) - pow(y,5), 0.2)) ); z>=1;z--){
				for(long w = min(z,  floor( pow( target - pow(x,5)-pow(y,5)-pow(z,5), 0.2) )); w>=1; w--){
					 cout<<"Checking "<<x<<" "<<" y "<< y << " z "<< z <<" w "<< w <<endl;
					 if( pow(w,5)+pow(z,5)+pow(y,5)+pow(x,5) == target ){
						 cout << "Success! Euler disproved"<<endl;
						 return;
					 }
				}
			}
		}
	}
    return;

}

int main()
{
	disprove_euler(144);
	return 0;
}

Interestingly, Kostis Gourgoulias tweeted about another paper, that contains fewer words.

conway.png

However, the shortest proof of an important result in my opinion is this one, even if there were no mathematical journals in Pythagoras’ time.

pythagoras7.gif

Do you know any short proofs of recent mathematical results?

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