Unsolved Problems

Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this ...

Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)....

Every even integer greater than 2 can be expressed as the sum of two primes....

Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?...

Are there infinitely many primes of the form $n^2 + 1$?...

The only solution to $x^p - y^q = 1$ in natural numbers x, y > 0 and p, q > 1 is $3^2 - 2^3 = 1$....

If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...

Extend the theory of quadratic forms with algebraic numerical coefficients....

Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$....

For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?...

Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?...

How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?...

Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?...

For any $\varepsilon > 0$, is there a constant $c(\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\varepsilon) x^{1/2-\varepsilon}$?...

If Euler's totient function $\phi(n)$ divides $n-1$, must $n$ be prime?...

Does there exist a 3×3 magic square composed entirely of distinct perfect squares?...

Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?...

Does the partition function satisfy any arbitrary congruence infinitely often?...

Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?...

Are there infinitely many perfect numbers?...

Do quasiperfect numbers exist?...

Do any almost perfect numbers exist that are not powers of 2?...

Are there exactly 65 idoneal numbers, or could there be 66 or 67?...

Do any pairs of amicable numbers exist where one is odd and one is even?...

Are there infinitely many pairs of amicable numbers?...

Are there infinitely many Giuga numbers?...

Do any odd weird numbers exist?...

Does there exist a covering system of congruences using only odd distinct moduli?...

If $B$ is an additive basis of order 2, must the representation function tend to infinity?...

If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \geq k$?...

Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?...

Can an algorithm determine if a constant-recursive sequence contains a zero?...

Do the Ulam numbers have a positive density?...

Can integer factorization be solved in polynomial time on a classical computer?...

Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?...

Is $p$ prime if and only if $pB_{p-1} \equiv -1 \pmod{p}$ for the Bernoulli number $B_{p-1}$?...

Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?...

Do quasiperfect numbers exist?...

Do odd weird numbers exist?...

Are there infinitely many pairs of amicable numbers?...

Does iterating unsigned differences on prime sequence always yield 1 as first element?...

If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?...