Unsolved Problems

Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For...

For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$....

Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?...

Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1....

Find all integer solutions to $n! + 1 = m^2$....

For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z....

For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$?...

Do Lychrel numbers exist in base 10?...

Is 10 a solitary number (no other number shares its abundancy index)?...

Does every nonnegative integer appear in Recamán's sequence?...

Do Lychrel numbers exist in base 10?...