Problem Workshop

For any positive integer n, iterate f(n) = n/2 if n is even, f(n) = 3n+1 if n is odd. Does the sequence always reach 1?

Are there infinitely many primes p such that p+2 is also prime? Examples: (3,5), (11,13), (17,19), \ldots

Can every even integer n > 2 be written as the sum of two primes? E.g., 10 = 3+7 = 5+5, 100 has 6 representations.

A *perfect number* equals the sum of its proper divisors: 6, 28, 496, 8128, \ldots Every known perfect number is even. Does an *odd* perfect…

If A^x + B^y = C^z in positive integers with x, y, z > 2, then A, B, C share a common prime factor. A generalization of Fermat's Last Theore…

For every integer n \ge 2, can 4/n be written as 1/a + 1/b + 1/c with positive integers a, b, c? E.g., 4/5 = 1/2 + 1/4 + 1/20.

k+1 runners on a unit circular track start at the same point with distinct constant speeds. For every runner, does there exist a time when t…

Is the number of occurrences of N > 1 in Pascal's triangle bounded by an absolute constant K? Most integers appear exactly twice; 120 appear…

Aliquot sequence: start at n, iterate s(n) = (sum of proper divisors). Terminates at 0, enters a perfect/amicable/sociable cycle, or diverge…

Start with a(0)=0. At step n, first try subtracting n from the previous term. Use the result if it is nonnegative and has not appeared befor…

Find all natural-number solutions of 1^k + 2^k + \cdots + (m-1)^k = m^k. The only known solution is m=3, k=1 (1+2=3). Erdős conjectured this…

Find all positive integers (n, m) with n! + 1 = m^2. Three known: (4,5), (5,11), (7,71). Conjectured (Brocard 1876, Ramanujan 1913): no othe…

Is there a 3\times 3 magic square (rows, columns, diagonals all sum to the same value) whose nine entries are *distinct positive perfect squ…

For any continuous simple closed curve C in the plane, must there be a square with all four vertices on C? Posed by Toeplitz (1911), open fo…

Color every point of the plane \mathbb{R}^2 so that any two points at distance exactly 1 get different colors. What is the minimum number of…

For any finite family \mathcal{F} closed under union (and not just \{\emptyset\}), is there always an element contained in at least half the…

For an integer polynomial P(x) = a_n \prod (x - \alpha_i), the *Mahler measure* is M(P) = |a_n| \prod \max(1, |\alpha_i|). Kronecker showed …

Find the smallest integer ES(n) such that any ES(n) points in general position in the plane contain n points forming a convex polygon. Known…

Does \sqrt{p_{n+1}} - \sqrt{p_n} < 1 hold for every n \ge 1, where p_n is the nth prime?

Does there exist a rectangular cuboid whose edges a, b, c, three face diagonals \sqrt{a^2+b^2}, \sqrt{a^2+c^2}, \sqrt{b^2+c^2}, and space di…

For any bounded convex polytope of dimension d with n facets, is its graph diameter bounded by a polynomial in n and d? The original Hirsch …

For n points in the plane, what is the minimum number of distinct distances g(n) they can determine? Erdős (1946) conjectured g(n) = \Omega(…

A Kakeya set in \mathbb{R}^n contains a unit segment in every direction. Besicovitch (1928) showed such sets can have Lebesgue measure zero.…

For any sequence x_1, x_2, \ldots \in \{+1, -1\}, do the partial sums S(n, d) = \sum_{k=1}^n x_{kd} always become arbitrarily large? Erdős (…

For n points in the plane, what is the maximum number u(n) of pairs at exactly unit distance? Erdős (1946) showed u(n) \ge n^{1 + c/\log\log…