Unsolved Problems

There are many unsolved problems in mathematics. The following is an illustrative, rather than exhaustive, selection of prominent problems and less widely known examples.

The Goldbach conjecture.

The Riemann hypothesis.

The conjecture that a Hadamard matrix exists for every positive order divisible by 4.

The twin prime conjecture. The bounded-gap results of Zhang and Maynard-Tao show that infinitely many pairs of primes have bounded separation, but do not establish separation 2 (Maynard 2015).

The P versus NP problem.

The Collatz problem. Tao (2022) proved that almost every Collatz orbit, in the sense of logarithmic density, eventually falls below any function tending to infinity, but this does not prove the conjecture for every starting value.

Determining whether the 196-algorithm fails to produce a palindromic number when started at 196.

Determining whether 10 is a solitary number. No friend of 10 is known, but the existence of one has not been ruled out (OEIS Foundation 2026).

Finding a general formula for the probability that two independent uniformly chosen elements generate the symmetric group . Dixon (1969) proved that this probability tends to , and stronger asymptotic estimates are known (Eberhard and Virchow 2019).

The happy end problem: determining the least such that every set of points in general position in the plane contains the vertices of a convex -gon. The conjecture is open in general; the best known asymptotic upper bound is (Holmsen et al. 2020).

Finding a perfect cuboid, i.e., an Euler brick whose space diagonal is also an integer.

Determining which integers can be written as a sum of three positive or negative cubic numbers. The cases 33 and 42, long the smallest unresolved admissible integers, were represented in 2019 (Booker 2019, Booker and Sutherland 2021), but the general problem remains open.

Determining which integers can be written as a sum of four positive or negative cubic numbers.

Lehmer's Mahler measure problem.

Lehmer's totient problem: determining whether a composite number can satisfy , where is the totient function.

Determining whether the Euler-Mascheroni constant is irrational.

Finding an exact analytic form for the square-site percolation threshold.

Determining whether any odd perfect numbers exist. Such a number, if it exists, is greater than (Ochem and Rao 2012).

The seven Millennium Prize Problems form a separate named collection. Six remain unsolved, while the Poincaré conjecture was proved by Perelman (Clay Mathematics Institute 2026).

In 1900, David Hilbert proposed a list of 23 outstanding problems in mathematics (Hilbert's problems), a number of which have now been solved, but some of which remain open. In 1912, Landau proposed four simply stated problems, now known as Landau's problems, which continue to defy attack even today. One hundred years after Hilbert, Smale proposed a list of 18 outstanding problems (Smale's problems).

The Open Problems Project and collections by Eppstein, Finch, Kimberling, and West provide further examples. Classic texts on unsolved problems in various areas of mathematics are Croft et al. (1991), in geometry, and Guy (2004), in number theory.