Unsolved problems in mathematics are numerous. The following is an illustrative, rather than exhaustive, selection of prominent problems and less
widely known examples.
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 Collatz conjecture. Tao (2022) proved that, for every function
tending to infinity, the Collatz orbit starting at contains a value at most for almost all positive
integers,
in the sense of logarithmic density. This does not prove the conjecture for every
starting value.
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).
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).
Determining which integers can be written as a sum of three cubes of integers.
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 whether any odd perfect numbers exist. Such a number, if it exists, is greater than (Ochem and Rao 2012).
For two independent uniformly chosen elements of the symmetric group, Dixon (1969) proved that the probability
of generating
tends to
as
tends to infinity. Stronger asymptotic estimates are known (Eberhard and Virchow
2019).
In 1900, David Hilbert (1902) 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. Landau (1912, p. 224) proposed
four simply stated problems, now known as Landau's
problems, which continue to defy attack even today. One hundred years after Hilbert,
Smale (1998) 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.
tending to infinity, the Collatz orbit starting at 
contains a value at most 
for almost all positive
integers 
,
in the sense of logarithmic density. This does not prove the conjecture for every
starting value.
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).
can satisfy 
, where 
is the totient function.
(Ochem and Rao 2012).
, Dixon (1969) proved that the probability
of generating 
tends to 
as 
tends to infinity. Stronger asymptotic estimates are known (Eberhard and Virchow
2019).
."
Math. Comput. 81, 1869-1877, 2012.