A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, 1 + 2 + 3 = 6 implies 6 is a perfect number.
Unsolved Problem: Are there any odd perfect numbers?
The belief, by some, that there are none goes back over 500 years (wikipedia).
If you want to check out some recent research into this problem, see oddperfect.org.
(Another unsolved problem: Are there an infinite number of even perfect numbers?)
In 1911, Otto Toeplitz asked the following question.
Inscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square?
This question, also known as the square peg problem or the Toeplitz’ conjecture, is still unsolved in general. (It is known in lots of special cases.)
Thanks to Mark Meyerson (“Equilateral triangles and continuous curves”,Fundamenta Mathematicae, 1980) and others, the analog for triangles is true. For any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C. (In particular, the triangle can be equilateral.) The survey page by Mark J. Nielsen has more information on this problem.
Added 2016-11-23: See also this recent post by T. Tao.
Added 2020-07-01: This has apparently been solved by Joshua Greene and Andrew Lobb! See their ArXiV paper (https://arxiv.org/abs/2005.09193).
In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.
First, we define the function on the set of positive integers:
If the number is even, divide it by two: .
If the number is odd, triple it and add one: .
In modular arithmetic notation, define the function as follows:
, and . Believe it or not, this is the restriction to the positive integers of the complex-valued map .
The 3n+1 conjecture is: The sequence
will eventually reach the number 1, regardless of which positive integer is chosen initially.
This is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.