Concept

Open problem

A question that has been stated precisely, attacked seriously, and not settled. The useful ones come with an identified obstruction — a named reason the available methods stop short — rather than merely an absence of answers.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

dynamics · Collatz

Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes

Named alongside it

The objects these essays reach for when they reach for this one.

Heuristic Asymptotic Collatz Conjecture Counting argument Divergence Iteration Logarithm Orbit Parity Prime Termination

All concepts