Examples of Open Problems Solved by Short Proofs

A short proof can resolve a problem that resisted solution, but the phrase needs care. Sometimes the result is a famous conjecture that falls to one counterexample; sometimes only a small case is settled; and sometimes a short proof comes years after a much longer first proof. The examples below show these different meanings.

A counterexample to Euler’s sum-of-powers conjecture

Euler conjectured that, for an integer $k>2$, a positive $k$th power cannot be expressed as a sum of fewer than $k$ positive $k$th powers. The conjecture resembles Fermat’s Last Theorem, but allows more than two terms. It is false: four fifth powers already sum to a fifth power:

$$27^5+84^5+110^5+133^5=144^5.$$

For example, evaluating both sides gives

$$61{,}917{,}364{,}224=61{,}917{,}364{,}224.$$

That identity disproves the conjecture for exponent $5$ in a single line. Lander and Parkin reported the counterexample in 1966. The distinction is that checking the answer is easy, while finding such an identity is not: the compact disproof does not record the search that produced it. Nor does this one example settle every version of the broader sum-of-powers question.

The five-point happy ending problem

The happy ending problem asks when a set of points in the plane must contain points that are the vertices of a convex polygon. Its smallest nontrivial case says that any five points in general position—that is, no three on one line—contain four that form a convex quadrilateral.

Here is a short proof. Join every pair of the five points by a straight segment. If no two segments cross except at shared endpoints, these ten segments would draw the complete graph on five vertices without crossings. But a simple planar graph with five vertices has at most $3(5)-6=9$ edges. Since there are ten segments, a crossing must occur. The crossing segments have four distinct endpoints, and those four points form a convex quadrilateral: the crossing segments are its diagonals.

This is a compact counting proof of the five-point case. The problem was introduced by Esther Klein and developed by Paul Erdős and George Szekeres; the broader family of questions concerns how many points guarantee a convex $n$-gon. A short proof of the smallest case should not be mistaken for a short solution to every case.

A shorter proof of the prime number theorem

The prime number theorem states that the number of primes at most $x$, denoted by $\pi(x)$, satisfies

$$\pi(x)\sim \frac{x}{\log x}$$

as $x$ grows. In plain terms, among the integers up to $x$, the proportion that are prime is approximately $1/\log x$. Hadamard and de la Vallée Poussin proved the theorem independently in 1896 using substantial analytic arguments. In 1980, Donald J. Newman published a notably short analytic proof.

This is a different kind of example: Newman did not make the original open problem disappear with a one-line argument. He gave a shorter proof of a major result that already had a proof. It still uses complex analysis, so “short” here means short compared with the earlier treatment, not elementary or effortless for every reader.

What these examples do—and do not—show

A proof’s length is only one part of a mathematical achievement. A counterexample can be brief once discovered; a short special-case proof may leave the general problem open; and a later proof can compress a result without changing when it was first solved. In each case, check the exact statement and the scope of the argument before describing an open problem as solved.

There are also classic problems that are often presented as puzzles rather than as major research conjectures. For instance, to show there are arbitrarily long gaps between primes, take an integer $n\ge 2$. Each of

$$n!+2,\;n!+3,\;\ldots,\;n!+n$$

is composite, since $n!+j$ is divisible by $j$ for every $j$ from $2$ to $n$. This leaves a run of $n-1$ consecutive composite integers, so prime gaps have no fixed upper bound. The proof is short and useful, but its modern classroom presentation is not by itself evidence that a famous historical research problem was settled in one stroke.

Common mistakes

• Treating a counterexample as a proof of a positive claim. The fifth-power identity disproves Euler’s conjecture; it does not prove a general theorem about sums of powers.
• Confusing one case with the full problem. The five-point argument proves the smallest happy ending case, not every statement about convex polygons.
• Equating a short published proof with easy discovery. A polished argument may omit extensive searching, failed approaches, or earlier work.
• Calling a later proof the original solution. Newman’s proof of the prime number theorem is a shorter later proof, not the proof that first settled the theorem.