Unsolved Problems

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

Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?...

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

Do any odd perfect numbers exist?...

Are there infinitely many perfect numbers?...

Do quasiperfect numbers exist?...

Do Lychrel numbers exist in base 10?...

Do odd weird numbers exist?...

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

Is π a normal number (all digits equally frequent in all bases)?...

Are all irrational algebraic numbers normal?...

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?...

Are there infinitely many real quadratic fields with class number 1 (unique factorization)?...

Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields....

Does the p-adic regulator of an algebraic number field never vanish?...

Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?...

For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?...

Is the Euler-Mascheroni constant γ irrational? Transcendental?...

For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?...

If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?...

Can integer factorization be done in polynomial time?...

Let $a_1, \ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \ldots, n + a_k$ are all prime, p...

For all integers $x, y \geq 2$, we have $\pi(x+y) \leq \pi(x) + \pi(y)$, where $\pi(n)$ denotes the prime counting function (the number of primes less...

For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\gcd(a,b,c) = 1$, and discriminant $\Delta = b^2 - 4ac$ not a perfect square, the polynomial ta...

Let $\pi(n; a, b)$ be the number of primes $p \le n$ with $p \equiv a \pmod b$. For every $a$ and $b$ with $a \perp b$, are there infinitely many valu...

Let $\{a_i\}$ be any infinite sequence of integers for which $\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progress...

Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes...

Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime....

Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?...

Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \ln n \ln \ln n \ln \ln \ln \ln n / (\ln \ln \ln n)^2$ for arbitrarily large constan...

Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?...

For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?...

Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it ...

Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - ...

Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity?...

Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \le i \le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0...

What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\prod_{i=1}^{k_1} (a+i) \equiv \prod_{i=1}^{k_2} (a+k_1+i) \equiv \p...

Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?...

Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \dots + (n-1)^{n-1} + 1$, then $n$ is prime?...

In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of...

Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \le 2^k < n$?...

What is the least positive odd number not of the form $\pm p^a \pm 2^b$, where $p$ is an odd prime?...

Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the dia...

Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?...

Does there exist a composite number $n \equiv 3$ or $7 \pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?...

Does there exist an even Fibonacci pseudoprime?...

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $ N \gg ...

If $A\subseteq \mathbb{N}$ has $\sum_{n\in A}\frac{1}{n}=\infty$ then must $A$ contain arbitrarily long arithmetic progressions?...

Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $ \lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C? $ ...